# Shell-model study of titanium isotopic chain with chiral two- and three-body
forces

L. Coraggio  Dipartimento di Matematica e Fisica, Universit$\grave{a}$ degli
Studi della Campania ”Luigi Vanvitelli”, viale Abramo Lincoln 5, I-81100
Caserta, Italy  Istituto Nazionale di Fisica Nucleare, Complesso Universitario
di Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy  G. De Gregorio
Dipartimento di Matematica e Fisica, Universit$\grave{a}$ degli Studi della
Campania ”Luigi Vanvitelli”, viale Abramo Lincoln 5, I-81100 Caserta, Italy
Istituto Nazionale di Fisica Nucleare, Complesso Universitario di Monte S.
Angelo, Via Cintia, I-80126 Napoli, Italy  A. Gargano  Istituto Nazionale di
Fisica Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia,
I-80126 Napoli, Italy  N. Itaco  Dipartimento di Matematica e Fisica,
Universit$\grave{a}$ degli Studi della Campania ”Luigi Vanvitelli”, viale
Abramo Lincoln 5, I-81100 Caserta, Italy  Istituto Nazionale di Fisica
Nucleare, Complesso Universitario di Monte S. Angelo, Via Cintia, I-80126
Napoli, Italy  T. Fukui  RIKEN Nishina Center, Wako 351-0198, Japan  Y. Z. Ma
Guangdong Provincial Key Laboratory of Nuclear Science, Institute of Quantum
Matter, South China Normal University, Guangzhou 510006, China  F. R. Xu
School of Physics, and State Key Laboratory of Nuclear Physics and Technology,
Peking University, Beijing 100871, China

###### Abstract

The even-even Ti isotopic chain, from $A=42$ to 70, has been studied within
the nuclear shell-model framework by employing an effective Hamiltonian which
is derived by way of many-body perturbation theory from a chiral potential
with two- and three-body forces, and includes three-body contributions which
account for Pauli principle violations in nuclei with more than two valence
particles. We consider 40Ca as a closed core and a model space spanned by the
neutron and proton $0f1p$ orbitals with the addition of the $0g_{9/2}$ orbital
for neutrons. Calculated two-neutron separation energies and excitation
energies of the yrast $2^{+}$ states are reported and compared with the
experimental data, which are available up to 62Ti. The present study intends
to investigate the effects of the adopted effective interactions on the
evolution of the shell structure.

##  I Introduction

In the last two decades or so, a large amount of data provided by radioactive
ion beam facilities has refined our vision of the nuclear shell structure. The
existence of the magic numbers associated with 2, 8, 20, 28, 50, 82 protons or
neutrons, and 126 neutrons has been the cornerstone of the shell model for
many years starting from its entry into nuclear physics by Mayer and Jensen
Mayer (1949); Haxel et al. (1949). The new data have put into evidence that
magic numbers are not universal, but they may change as a function of $N$ and
$Z$ when going far from the stability line. For instance, new shell gaps have
been identified at $N=32,34$ in neutron-rich nuclei with $Z\approx 20$ and at
$N=16$ for O isotopes, while the disappearance or weakening of canonical magic
numbers has been observed in lighter nuclei at $N=8$, $N=20$, and $N=28$ (see,
for instance, Refs.Huck et al. (1985); Motobayashi et al. (1995); Navin et al.
(2000); Iwasaki et al. (2000); Janssens et al. (2002); Shimoura et al. (2003);
Dinca et al. (2005); Bürger et al. (2005); Gade et al. (2006); Bastin et al.
(2007); Hoffman et al. (2009); Kanungo et al. (2009); Takeuchi et al. (2012);
Wienholtz et al. (2013); Steppenbeck et al. (2013); Liu et al. (2019)).
Nowadays, great theoretical and experimental efforts are devoted to study the
evolution of the shell structure and, in particular, to identify the behavior
of shell gaps along isotopic chains as well as to understand the role played
by the different components of the nuclear force in driving their
modifications.

In particular, the Ca isotopic chain, with magic proton number $Z=20$, has
attracted considerable interest, since it exhibits two neutron shell closures
at $N=32$ Huck et al. (1985); Gade et al. (2006); Wienholtz et al. (2013) and
34 Steppenbeck et al. (2013) in addition to the standard one at $N=28$.
Special attention is presently focused on the very exotic 60Ca with the aim to
acquire information useful to investigate the shell structure evolution at
$N=40$ and shed light on the possible doubly magic nature of this nucleus.
From the experimental point of view the existence of 60Ca has been only
recently established Tarasov et al. (2018), but no information is available on
the excitation energy of the yrast $2^{+}$ state. It is worth mentioning also
that the experimental location of the neutron drip line in Ca isotopes is
still unknown, and no clear indications emerge from theory Meng et al. (2002);
Holt et al. (2012); Hagen et al. (2012); Hergert et al. (2014); Neufcourt et
al. (2019); Cao et al. (2019); Li et al. (2020).

On the other hand, the shell closure at $N=40$ observed in 68Ni, which
corresponds to the filling of the neutron $0f1p$ and proton $0f_{7/2}$
orbitals, rapidly disappears when removing protons from $0f_{7/2}$, as it is
shown by the behavior of the experimental excitation energies of the yrast
$J^{\pi}=2^{+}$ states in iron and chromium neutron-rich isotopes.

In this context, also the chain of Ti isotopes ($Z=22$) represents a key piece
to investigate the evolution of the shell structure towards $Z=20$. The
properties of these nuclei, especially those for $N\approx 40$, may provide a
more sound testing ground of nuclear models and of their predictions of the Ca
drip line.

Experimental data are now known for Ti isotopes from $A=42$ to 62. The masses
of 60,62Ti were determined for the first time through the measurements
reported in Ref. Michimasa et al. (2020), while the $2^{+}_{1}\rightarrow
0^{+}_{gs}$ and $4^{+}_{1}\rightarrow 2^{+}_{1}$ transitions in 60Ti and 62Ti
were observed via one-proton stripping reactions from 61V and 63V,
respectively, in Refs. Gade et al. (2014); Cortès et al. (2020). These
experiments show an increase in the $2^{+}_{1}$ excitation energies of both
nuclei as compared with the corresponding Cr and Fe isotones, which indicates
a decrease in collectivity towards $Z=20$. However, the reduced size in the
increase does not seem to support the existence of a shell closure at $N=40$
for 60Ca.

From the theoretical point of view, several shell-model (SM) calculations have
been carried out for Ti nuclei by considering various model spaces referred to
both 40Ca or 48Ca as closed core and employing empirically corrected
interactions Poves et al. (2005); Honma et al. (2005); Dinca et al. (2005);
Lenzi et al. (2010); Suzuki et al. (2013); Wimmer et al. (2019); Gade et al.
(2014); Cortès et al. (2020). In Ref. Coraggio et al. (2014), some of the
present authors performed a study of isotopic chains ”north-east” of 48Ca,
including Ti isotopes, by employing two-body matrix elements of the residual
interaction derived from the high-precision CD-Bonn free nucleon-nucleon
potential.

In the present paper, the whole even-even titanium isotopic chain from $A=42$
to 70 has been studied for the first time within the realistic SM framework
starting from up-to-date two- and three-body forces derived in the framework
of chiral perturbation theory (ChPT). The SM effective Hamiltonian, consisting
of the single-particle (SP) energies and the two-body matrix elements (TBMEs),
is built up using the many-body perturbation theory Kuo et al. (1971);
Coraggio and Itaco (2020), and includes the effect of second-order three-body
diagrams arising in nuclei with more than two valence particles, which account
for three-body correlations induced by the interaction via the two-body force
of clusters of three-valence nucleons with core excitations as well as with
virtual intermediate nucleons scattered above the model space.

This work follows two of our previous studies. In the first one Ma et al.
(2019), we investigated the role played by genuine and effective chiral three-
body forces in providing a reliable monopole component of the SM effective
Hamiltonian defined in the $0f1p$ model space. Results are discussed for
$Z=20$, 22, 24, 26, 28 with $N$ from 22 to 40. In the second one Coraggio et
al. (2020), focused only on Ca isotopes, the same approach was used in a
larger model space including the neutron $0g_{9/2}$ orbital to describe
isotopes beyond $N=40$. Here, we move a step forward to test our approach
along an extensive isotopic chain, as Ti nuclei, with both valence neutrons
and protons.

It is worth mentioning that two- and three-body chiral forces have been also
used in some recent ab initio calculations for Ti isotopes. In Ref. Hagen et
al. (2012), the authors investigated 50,54,56Ti by way of the coupled-cluster
method including the coupling to the particle continuum, while in Ref. Xu et
al. (2019) the shell gap in 54Ti was analyzed in terms of valence-space in-
medium similarity renormalization group (VS-IMSRG). The latter was also used
in Ref. Stroberg et al. (2021) to investigate the properties of nuclei from
helium to iron. Precision mass measurements of 51-55Ti were performed in Ref.
Leistenschneider et al. (2018) and compared with the predictions of
multireference in-medium similarity renormalization group (MR-IMSRG), VS-
IMSRG, and self-consistent Gorkov-Green’s function (GGF) calculations.
Furthermore, VS-IMSRG results are reported in Ref. Cortès et al. (2020) for
the excitation energies of the new observed $2^{+}$ and $4^{+}$ states in
62Ti, together with predictions of beyond mean-field and large-scale SM
calculations, the latter being based on the same model space and effective
interaction of Ref. Lenzi et al. (2010). The comparison with experiment shows
that only large-scale SM calculations reproduce very accurately data for 62Ti,
which are instead largely overestimated by the other two approaches. The
theoretical framework of Ref. Lenzi et al. (2010) gives also a good
description of the experimental excitation energies of 60Ti, as shown in Ref.
Gade et al. (2014).

In concluding this section, it has to be pointed out that, similarly to our
previous study on Ca isotopes Coraggio et al. (2020), we do not include in the
model space the neutron $1d_{5/2}$ orbital, which was shown to be of great
importance to reproduce the onset of the collectivity at $N=40$ for isotopic
chains ”north-east” of 48Ca, as first discussed in Ref. Caurier et al. (2002)
and then confirmed in Refs. Lenzi et al. (2010); Coraggio et al. (2014);
Wimmer et al. (2019); Gade et al. (2014); Cortès et al. (2020).

In Ref. Zuker et al. (1995), the authors have identified the source of this
collective behavior as a consequence of the quasi-SU(3) approximate symmetry,
owing to the interplay between the quadrupole-quadrupole component of the
residual interaction and the central field in the sub-space spanned by the
lowest $\Delta j$=2 orbitals of a major shell.

In particular, in Ref. Lenzi et al. (2010) -where the adopted model space was
based on a 48Ca core including the $1p0f$ shell for protons and the
$1p,0f_{5/2},0g_{9/2},1d_{5/2}$ orbitals for neutrons - the deformation
driving role of the neutron $1d_{5/2}$ orbital below 68Ni was assessed showing
that the maximum deformation develops in 64Cr and decreases towards Ca. Such a
situation was, in fact, explained in terms of a reduction of the neutron
$0f_{5/2}-0g_{9/2}$ gap when protons are removed from the $0f_{7/2}$ orbital,
which is accompanied by an enhancement of the quadrupole-quadrupole
correlations between the neutron $0g_{9/2}$ and $1d_{5/2}$ orbitals. Similar
results are shown in Ref. Coraggio et al. (2014) where the Ca, Ti, Fe Cr, Ni
isotopic chains ”north-east” of 48Ca were studied within the realistic SM
framework, by comparing the results for two model spaces differing for the
inclusion of the neutron $1d_{5/2}$ orbital.

A model space space starting from 40Ca as closed core and including the
neutron $1d_{5/2}$ makes calculations around $N=40$ very cumbersome.
Dimensions of the Hamiltonian matrix reach about $10^{11}$ in the case of
62Ti, which represents the limit of our computing power. Our choice to exclude
the neutron $1d_{5/2}$ orbital is, therefore, related essentially to this
reason. We plan to overcome this limit by extending the so-called double-step
procedure introduced in Refs. Coraggio et al. (2015, 2016) to reduce the
computational complexity of large-scale SM calculations, which consists in
deriving an effective Hamiltonian within a manageable model space by means of
a unitary transformation of a large-scale Hamiltonian.

Nevertheless, the present study, in parallel with our previous one on Ca
isotopes, intends to investigate the effects of our SM effective interactions
on the evolution of the shell structure before tackling the problem in a
larger model space including the neutron $1d_{5/2}$ orbital.

In the following section, we give an outline of the theoretical framework in
which our SM calculations are performed. Results for the excitation energies
of the yrast $2^{+}$ states and two-neutron separation energies of even-even
Ti nuclei from $N=20$ to 48 are presented and compared with the available
experimental data in Sec. III. In this section, we also discuss the
sensitivity of these results with respect to the many-body correlations and
their impact on the effective-single particle energies. Section IV provides a
summary and concluding remarks.

##  II Outline of the theoretical framework

As mentioned in the Introduction, SM calculations for Ti isotopes have been
performed within the same approach as Ref. Coraggio et al. (2020), which we
refer for more details. Calculations are performed by means of the SM code
KSHELL Shimizu et al. (2019) in the model space spanned by the four proton
$0f_{7/2}$, $0f_{5/2}$, $1p_{3/2}$, $1p_{1/2}$ orbitals and the five neutron
$0f_{7/2}$, $0f_{5/2}$, $1p_{3/2}$, $1p_{1/2}$, $0g_{9/2}$ orbitals outside
the doubly magic 40Ca.

We start our calculation from the nucleon-nucleon ($NN$) potential developed
by Entem and Machleidt Entem and Machleidt (2002) within chiral perturbation
theory at next-to-next-to-next-to-leading order (N3LO) and the chiral three-
body ($NNN$) potential at next-to-next-to-leading order (N2LO), which share
the same nonlocal regulator function. The low-energy constants (LECs)
appearing in both $NN$ and $NNN$ components, namely $c_{1}$, $c_{3}$, $c_{4}$,
are determined by the renormalization procedure described in Ref. Machleidt
and Entem (2011), while for the $c_{D}$ and $c_{E}$ LECs characterizing only
the $NNN$ force we take the values of Ref. Navrátil et al. (2007).

The matrix elements of the $NN$ and $NNN$ forces, with the addition of the
Coulomb one in the proton-proton channel, are computed in the harmonic
oscillator basis with an oscillator parameter
$\hbar\omega=45A^{-1/3}-25A^{-2/3}$ for $A=40$. Details of the calculation of
matrix elements of the N2LO $NNN$ potential are reported in Ref. Fukui et al.
(2018).

These matrix elements are used as input to derive the SM effective Hamiltonian
$H_{\rm eff}$ within the time-dependent perturbation theory. Specifically,
$H_{\rm eff}$ is expressed by way of the Kuo-Lee-Ratcliff folded-diagram
expansion Kuo et al. (1971) in terms of the $\hat{Q}$-box vertex function,
which is defined as

$\hat{Q}(\epsilon)=PH_{1}P+PH_{1}Q\frac{1}{\epsilon-QHQ}QH_{1}P~{},$ (1)

where $H$ is the full nuclear Hamiltonian $H=H_{0}+H_{1}$, $H_{0}$ and $H_{1}$
being the unperturbed and the interaction components, respectively, and
$\epsilon$ an energy parameter called starting energy”.

Then, the $\hat{Q}$-box may be calculated by expanding the term $1/(\epsilon-
QHQ)$ as a power series

$\frac{1}{\epsilon-QHQ}=\sum_{n=0}^{\infty}\frac{1}{\epsilon-
QH_{0}Q}\left(\frac{QH_{1}Q}{\epsilon-QH_{0}Q}\right)^{n}~{}.$ (2)

This provides a perturbative expansion of the $\hat{Q}$-box, and its
diagrammatic representation is given as a collection of irreducible valence-
linked Goldstone diagrams Kuo et al. (1971).

Once the $\hat{Q}$-box is calculated, $H_{\rm eff}$ is obtained solving
nonlinear matrix equations by way of iterative techniques such as the Kuo-
Krenciglowa and Lee-Suzuki ones Suzuki and Lee (1980), or graphical
noniterative methods Suzuki et al. (2011). The latter is the method we have
employed in present work, since it results in a faster and more stable
convergence to the solution of the matrix equations.

We arrest the $\hat{Q}$-box expansion of the one- and two-body Goldstone
diagrams at third order in the $NN$ potential and at first order in the $NNN$
one. It has to be pointed out that the diagrams at first order in $NNN$
potential - whose analytical expressions are reported in Refs. Fukui et al.
(2018); Ma et al. (2019) \- are the coefficients of the one-body and two-body
terms arising from the normal-ordering decomposition of the three-body
component of a many-body Hamiltonian Hjorth-Jensen et al. (2017).

Since our goal is the study of the Ti isotopic chain up to 70Ti, we are going
to diagonalize the SM Hamiltonian for systems up to thirty valence nucleons.
This means that the derivation of the SM effective Hamiltonian needs to
resolve the progressive filling of the model space orbitals, especially in the
calculation of the irreducible valence-linked diagrams of the $\hat{Q}$-box.

This should be achieved by including in the $\hat{Q}$-box many-body diagrams
which account for the interaction via the two-body force of the valence
nucleons with configurations outside the model space, leading to a dependence
of $H_{\rm eff}$ on the number of valence nucleons. We arrest the cluster
expansion to the leading term, namely second-order three-body diagrams, which,
for those nuclei with more than two valence nucleons, account for the
interaction of the valence nucleons with core excitations as well as with
virtual intermediate nucleons scattered above the model space. These diagrams
are reported in Fig. 1 and their explicit expressions, $D_{A}$ and $D_{B}$,
are

$\langle\left[(j_{a}j_{b})_{J_{ab}},j_{e}\right]_{J}|D_{A}|\left[(j_{c}j_{d})_{J_{cd}},j_{f}\right]_{J}\rangle=\\\
\sum_{J_{be}}\sum_{p}(-1)^{b+e+f+p}\hat{J_{ab}}\hat{J_{cd}}\hat{J_{be}}^{2}\left\\{\begin{array}[]{ccc}j_{a}~{}j_{b}~{}J_{ab}\\\
j_{e}~{}J~{}J_{be}\end{array}\right\\}\left\\{\begin{array}[]{ccc}j_{a}~{}j_{p}~{}J_{cd}\\\
j_{f}~{}J~{}J_{be}\end{array}\right\\}\\\ ~{}\times\frac{\langle
b,e;J_{be}|V_{NN}|p,f;J_{be}\rangle\langle
a,p;J_{cd}|V_{NN}|c,d;J_{cd}\rangle}{[\epsilon_{0}-(\epsilon_{a}+\epsilon_{f}+\epsilon_{p})]}~{},$
(3)

$\langle\left[(j_{a}j_{b})_{J_{ab}},j_{e}\right]_{J}|D_{B}|\left[(j_{c}j_{d})_{J_{cd}},j_{f}\right]_{J}\rangle=\\\
\sum_{J_{be}}\sum_{h}(-1)^{b+e+f+h+1}\hat{J_{ab}}\hat{J_{cd}}\hat{J_{be}}^{2}\left\\{\begin{array}[]{ccc}j_{a}~{}j_{b}~{}J_{ab}\\\
j_{e}~{}J~{}J_{be}\end{array}\right\\}\left\\{\begin{array}[]{ccc}j_{a}~{}j_{h}~{}J_{cd}\\\
j_{f}~{}J~{}J_{be}\end{array}\right\\}\\\ ~{}\times\frac{\langle
b,e;J_{be}|V_{NN}|h,f;J_{be}\rangle\langle
a,h;J_{cd}|V_{NN}|c,d;J_{cd}\rangle}{[\epsilon_{0}-(\epsilon_{a}+\epsilon_{b}+\epsilon_{d}+\epsilon_{f}-\epsilon_{h})]}~{},$
(4)

where the indices $a,b,c,d,e,f,p,h$ refer to the quantum numbers of the
incoming, outcoming, and intermediate single-particle states, $\epsilon_{m}$
denotes the unperturbed single-particle energy of the orbital $j_{m}$,
$\epsilon_{0}$ is the so-called starting energy, namely the unperturbed energy
of the incoming particles
$\epsilon_{0}=\epsilon_{c}+\epsilon_{d}+\epsilon_{f}$.

From now on, we dub these contributions 3-b correlations .

Figure 1: Second-order three-body diagrams. The sum over the intermediate
lines runs over particle and hole states outside the model space, shown by $A$
and $B$, respectively. We report only one of the nine existing topologies
which correspond to the permutations of the external lines.

We also point out that for each topology reported in Fig. 1, there are nine
diagrams, corresponding to the possible permutations of the external lines
Polls et al. (1983).

Diagrams $A,B$ include the effects of the Pauli blocking due to the filling of
the valence particle lines in the second-order ladder and core-polarization
diagrams Ellis and Osnes (1977), respectively, quenching the contribution of
these two-body terms.

Nevertheless, the KSHELL SM code, which we employ for the calculations, cannot
perform the diagonalization of a three-body $H_{\rm eff}$, so we derive
density-dependent two-body terms ($\alpha$) from the corresponding second-
order three-body diagrams ($A,B$), as shown in Fig. 2.

Figure 2: Density-dependent two-body contribution that is derived from a
three-body one. $\alpha$ is obtained by summing over one incoming and outgoing
particle of the three-body graphs $A,B$ reported in Fig. 1.

We then calculate, for each $A,B$ topology, nine one-loop diagrams, namely the
graph $(\alpha)$ in Fig. 2. Their explicit form, in terms of the three-body
graphs $A,B$, is

$\langle(j_{a}j_{b})_{J}|V^{\alpha}|(j_{c}j_{d})_{J}\rangle=\\\
\sum_{m,J^{\prime}}~{}\rho_{m}\frac{\hat{J^{\prime}}^{2}}{\hat{J}^{2}}\langle\left[(j_{a}j_{b})_{J},j_{m}\right]_{J^{\prime}}|V^{A,B}|\left[(j_{c}j_{d})_{J},j_{m}\right]_{J^{\prime}}\rangle~{}~{},$
(5)

where the summation over $m$-index runs in the model space, and $\rho_{m}$ is
the unperturbed occupation density of the orbital $m$ according to the number
of valence nucleons.

Following the above procedure, the perturbative expansion of the $\hat{Q}$-box
contains one- and two-body diagrams up to third order in $V_{NN}$, and a
density-dependent two-body contribution which accounts for three-body second-
order diagrams $A,B$.

It should be stressed that the latter term depends on the number of valence
protons and neutrons, thus we derive specific effective SM Hamiltonians for
any nuclear system under consideration, Hamiltonians that differ only for the
two-body matrix elements.

It is worth mentioning that, as done in Ref. Coraggio et al. (2020) for Ca, we
have checked the effect of the spurious center-of-mass motion. In line with
the previous outcome, we have found that results are only marginally affected
by spurious components.

In the Supplemental Material SM  we report the TBMEs of the interaction
calculated without including three-body correlations as well as those of the
density dependent interaction for $A=62$. Neutron and proton SP energies are
shown in Table 1. It is worth mentioning that the proton and neutron SP
spacings obtained from the theory are shifted to reproduce the experimental
ground-state energies of 41Sc and 41Ca with respect to 40Ca. This is because,
as discussed in Ref. Coraggio et al. (2020), our derivation of the effective
Hamiltonian with $N_{\rm max}=18$ does not provide convergent spectra for the
one-valence systems.

Table 1: Proton $\epsilon_{b}^{\pi}$ and neutron $\epsilon^{\nu}_{b}$ single-particle energies (in MeV). $b$ | $\epsilon^{\pi}_{b}$ | $\epsilon^{\nu}_{b}$
---|---|---
$0f_{7/2}$ | $-1.1$ | $-8.4$
$0f_{5/2}$ | $6.2$ | $-0.2$
$1p_{3/2}$ | $1.4$ | $-5.2$
$1p_{1/2}$ | $3.3$ | $-3.1$
$0g_{9/2}$ | $-$ | $1.5$

##  III Results

We start focusing on the two-neutron separation energies, whose calculated
values from 44Ti to 70Ti are shown in Fig. 3 together with the available
experimental data.

Figure 3: (Color online) Theoretical two-neutron separation energies for even-
even Ti isotopes from $N=22$ to 48, with and without the 3-b correlations, are
compared to the experimental data from Ref. Michimasa et al. (2020) for $N=40$
and from AME2020 Wang et al. (2021) for the other isotopes.

We see that the calculated two-neutron separation energies ($S_{2n}$), with
(magenta squares) and without (blue diamonds) 3-b correlations, are in a good
agreement with the experimental data (red cross) up to $N=32$. Actually, both
calculations provide the observed shell closures at $N=28$ and $N=32$, which
manifest as a sudden drop of $S_{2n}$ at $N=30$ and $N=34$. However, starting
from $N=34$ the effect of 3-b correlations becomes rather large and, as
expected, grows with increasing neutron number in analogy with the results
obtained for Ca isotopes Coraggio et al. (2020). It turns out that their
contribution increases the calculated $S_{2n}$, and leads to a good agreement
with the measured values in 56-62Ti, which are significantly underestimated
when 3-b correlations are omitted. Nevertheless, in both calculations all Ti
isotopes up to 70Ti are predicted to be bound.

The experimental excitation energies of the yrast $2^{+}$ state are compared
with the calculated values obtained with and without 3-b correlations in Fig.
4. Up to $N=38$, the results of the two calculations are very similar and do
not differ significantly from those reported in Ref. Ma et al. (2019), where
the model space was limited to the neutron and proton $1f1p$ orbitals. It can
be seen that both calculations are in quite good agreement with experimental
data up to $N=34$, and in particular predict the observed shell closures in
50Ti and 52Ti. However, from $N=36$ on, the theoretical curves start to climb
towards $N=40$, at variance with the observed behavior of the yrast $2^{+}$
excitation energies that does not exhibit such a raise. This deficiency in our
calculations may be ascribed to the inadequacy of the adopted model space,
which does not include the neutron $1d_{5/2}$ orbital. In fact, as mentioned
in the Introduction, the interaction between neutron $1d_{5/2}$ and $0g_{9/2}$
orbitals ignites a quadrupole collectivity that is responsible for the
disappearance of the $N=40$ shell closure in isotopic chains ”north-east” of
48Ca, especially in chromium and iron isotopes Caurier et al. (2002); Lenzi et
al. (2010); Coraggio et al. (2014); Wimmer et al. (2019); Gade et al. (2014);
Cortès et al. (2020).

Despite this unsatisfactory result, it is worth pointing out that differences
emerge in the two theoretical curves starting from $N=40$ to 48. As a matter
of fact, they evince that the contribution of 3-b correlations reduces the
excitation energy of the $2^{+}_{1}$ state, leading to the disappearance of
the shell gap in 62Ti, in agreement with experiment Cortès et al. (2020).

We have found that 3-b correlations also influence the calculation of the
yrast $4^{+}$ states. The predicted behavior of their excitation energies,
obtained by omitting these correlations, exhibits a positive slope from 60Ti
to 62Ti that is not experimentally observed. On the other hand, the
experimental behavior is reproduced when 3-b correlations are included,
although the calculated excitation energies underestimate the measured values
by about 500 KeV along the whole chain, except for 42Ti where a larger
discrepancy is noticed.

The results shown above evince the effects produced on the shell structure of
Ti isotopes when the evolution the SM Hamiltonian, as a function of the number
of valence nucleons, is microscopically taken into account. We are confident
that the combination of these correlations with the enlargement of the model
space including the neutron $1d_{5/2}$ orbital is the key to reproduce the
observed degree of collectivity in N$\approx$40 nuclei ”north-east” of 48Ca.

Figure 4: (Color online) Theoretical $2^{+}_{1}$ excitation energies for even-
even Ti isotopes from $N=20$ to 48, with and without the 3-b correlations, are
compared with the experimental data from Ref. Cortès et al. (2020) for $N=40$
and from Ref. NND  for the other isotopes.

The shell evolution can be analyzed by studying the behavior of the effective
single particle energies (ESPEs) along the isotopic chain, and thus we have
found it interesting to see how they are influenced by 3-b correlations.

The ESPEs are defined as

${\rm
ESPE}(a^{\tau})=\epsilon_{a}^{\tau}+\sum_{b\tau^{\prime}}{\bar{V}}_{ab}^{\tau\tau^{\prime}}n_{b}^{\tau^{\prime}},$
(6)

where $\tau$,$\tau^{\prime}$ stand for neutron or proton index and $a$,$b$ run
over all the valence orbitals. The quantities $\epsilon_{a}^{\tau}$ and
$n_{a}^{\tau}$ denote the SP energy and the ground-state occupation number of
the $a^{\tau}$ level, while ${\bar{V}}_{ab}^{\tau\tau^{\prime}}$ is the
monopole component of the two-body effective interaction $(V_{\rm eff})$
obtained by averaging on the projection of the total angular momentum and can
be written as

${\bar{V}}_{ab}^{\tau\tau^{\prime}}=\frac{\sum_{J}(2J+1)\langle
a^{\tau},b^{\tau^{\prime}}\mid V_{\rm eff}\mid
a^{\tau},b^{\tau^{\prime}}\rangle_{J}}{\sum_{J}(2J+1)},$ (7)

where $J$ runs over Pauli allowed values.

The proton ESPEs are not significantly affected by 3-b correlations. In both
calculations, the proton $0f^{\pi}_{7/2}$ orbital is well separated from the
other ones, and the $1p^{\pi}_{3/2}-0f^{\pi}_{7/2}$ gap increases towards
neutron rich Ti isotopes, which indicates the stability of the shell closure
at $Z=28$. The other SP states are close each other with the inversion of
$1p^{\pi}_{1/2}$ and $0f^{\pi}_{5/2}$ orbitals in correspondence of $N=40$.

Figure 5: (Color online) ESPE for neutrons in even-even Ti isotopes from
$N=20$ to 48 resulting from calculations (a) without and (b) with 3-b
correlations.

More interesting is the behavior of the ESPEs for neutrons. Figure 5(a),
presenting the neutron ESPEs obtained by omitting 3-b correlations, shows some
similarities with Fig. 5(b), where these correlations are included. In both
calculations, the neutron $0f_{7/2}$ orbital is well isolated from the others
along all the isotopic chain, and a fairly large spacing is observed between
the $1p_{3/2}$ and $1p_{1/2}$ orbitals up to $N=44$, which gives rise to the
subshell closure at $N=28$ and $N=32$, respectively. No shell closure is
instead predicted at $N=34$, corresponding to the filling of the
$1p^{\nu}_{1/2}$ orbital, whose separation energy from the $0f^{\nu}_{5/2}$
orbital decreases with increasing neutron number. In spite of these
similarities, however, we note that 3-b correlations introduce appreciable
changes in the neutron ESPEs, which explain the differences in the two-neutron
separation energies and in the $2^{+}_{1}$ excitation energies discussed
above. In particular, we see that all the neutron ESPEs show a more rapid and
almost continuous decrease as a function of $N$ when 3-b correlations are
taken into account, while the neglect of these correlations leads to a more
flat behavior or even to an increasing trend for the $1p^{\nu}_{3/2}$,
$1p^{\nu}_{1/2}$, and $0f^{\nu}_{5/2}$ orbitals from $N=30$ or $34$ to 40.
This explains the strong negative slope of the $S_{2n}$ curve when
calculations are performed without including 3-b correlations. Moreover, the
$0g^{\nu}_{9/2}-0f^{\nu}_{5/2}$ gap is affected by 3-b correlations which
reduce its value by $\approx 400$ keV at $N=40$. The larger gap we find when
3-b correlations are neglected leads to the increase in energy of the
$2^{+}_{1}$ state from 60Ti to 62Ti we predict in such a case.

Figure 6: (Color online) Neutron-neutron monopole matrix elements,
${\bar{V}}^{\nu\nu^{\prime}}_{ab}$, of the effective interactions with and
without 3-b correlations (see text for details).

The different behavior of the neutron ESPEs with and without 3-b correlations
is related to the changes these correlations produce in the monopole part of
the neutron-neutron interaction. Actually, the neutron-proton monopole
components, which also enter in the neutron ESPE definition, are not
significantly affected by 3-b correlations, and, additionally, they do not
play a relevant role since their contribution is limited by the reduced number
of valence protons. As concerns the neutron-neutron monopole components, we
find that 3-b correlations induce on the overall a larger attractiveness which
explains the difference in the behavior of the neutron ESPEs shown in Figs.
5(a) and 5(b). As an example, we compare in Fig. 6 the monopole matrix
elements of the neutron-neutron effective interaction without 3-b correlations
with those of the density-dependent interaction at $N=40$. It can be seen that
the only matrix element that acquire a repulsive component is
${\bar{V}}_{1p_{1/2}1p_{1/2}}^{\nu\nu}$.

Furthermore, the size of changes induced by 3-b correlations is orbital
dependent, which leads to modifications of the ESPE spacings. In particular,
${\bar{V}}_{0g_{9/2}b}^{\nu\nu}$ receive a larger attractive contribution with
respect to ${\bar{V}}_{0f_{5/2}b}^{\nu\nu}$ for any value of $b$, except for
$b=0g_{9/2}$. This gives rise to the reduced gap between the $0_{9/2}$ and
$0f_{5/2}$ ESPEs we find at $N=40$ when 3-b correlations are taken into
account.

To better asses the role of the monopole contributions arising from 3-b
correlations we have investigated the interplay of monopole and multipole
components. To this end, we have built a modified effective interaction for
each Ti isotope starting from the Hamiltonian without 3-b correlations and
replacing its monopole component with that arising from the density-dependent
Hamiltonian derived for the same nucleus. The obtained results are very close
to those of the original density-dependent Hamiltonians, showing, in
particular, that the behavior of the two-neutron separation energies and of
the $2^{+}_{1}$ excitation energies is essentially determined by the monopole
component.

##  IV Summary and conclusions

In this paper we have studied the even-even Ti isotopic chain from $A=42$ to
70 within the realistic SM framework starting from up-to-date two- and three-
body forces derived from ChPT. The one and two-body matrix components of the
SM effective Hamiltonian are derived by means of the many-body perturbation in
a model space spanned by the proton and neutron $0f1p$ orbitals plus the
neutron $0g_{9/2}$ orbital. Three-body contributions are also included through
density-dependent TBME consistently derived within a microscopic approach from
chiral forces. These correlations are essential to account for the Pauli
blocking effect due to the progressive filling of the model space orbitals in
nuclei with more than two valence nucleons.

As in our previous paper Coraggio et al. (2020), we have compared the
calculated two-neutron separation energies and the excitation energies of the
yrast $2^{+}$ state with the available experimental data.

We have found that 3-b correlations have a tiny impact on the two-neutron
separation energies up to $N=32$ but, starting from $N=34$ on, their role
becomes more relevant with the increasing number of valence neutrons. This
results in an upshift of the two-neutron separation energies that become very
close to the experimental ones up 62Ti beyond which no experimental
information is available. This makes us confident in our predictions for the
heaviest Ti isotopes, which we found to be bound up to $N=70$. In Ref.
Coraggio et al. (2020), we have shown that 3-b correlations are relevant also
to determine the drip line in Ca isotopes. In fact, without the inclusions of
these correlations the drip line of calcium isotopes is located at $N=40$,
while their attractive contribution shifts the last bound nucleus at least to
68Ca.

As concerns the excitation energies of the yrast $2^{+}$ states, we do not
obtain a satisfactory agreement with experiment from $N=36$ on, despite the
inclusion of 3-b correlations. As a matter of fact, the decreasing trend
observed from $N=36$ to $40$ is not reproduced even if correlations are taken
into account. This may be related to the omission of the neutron $1d_{5/2}$
orbital, which, as discussed above, is fundamental to reproduce the
collectivity of nuclei ”north-east” of 48Ca around a $N=40$, especially of
64Cr and 62Fe. Nevertheless, our calculations evince that contributions
arising from 3-b correlations affect the neutron-neutron monopole components
leading to a reduction of the excitation energy of the $2^{+}_{1}$ state and
in particular of the shell gap at 62Ti. A similar result was found for the
excitation energy of $2^{+}_{1}$ state in 60Ca, whose value is reduced by
their inclusion from $\sim 2.3$ to 1.7 MeV, which is quite small when compared
with the gaps observed in 52Ca and 48Ca, namely $\approx 2.5$ and $\approx
4.0$ MeV.

Despite the missing contribution coming from the neutron $1d_{5/2}$ orbital,
interesting conclusions can be drawn. In fact, the present study shows that
starting from $N=34$ the attractive contribution arising from 3-b correlations
impacts significantly on the shell structure evolution.

We plan in a near future to overcome the computational complexity related to
the large model by including the neutron $1d_{5/2}$ orbital so as to perform
large-scale SM calculations to study the collective behavior at $N=40$
starting from chiral two- and three-body forces and employing single-particle
energies and two-body matrix elements derived from many-body perturbation
theory.

###### Acknowledgements.

We acknowledge the CINECA award under the ISCRA initiative code HP10B51E4M and
through the INFN-CINECA agreement for the availability of high performance
computing resources and support. G. De Gregorio acknowledges the support by
the funding program VALERE of Università degli Studi della Campania Luigi
Vanvitelli . The National Natural Science Foundation of China under Grants No.
11835001, 11921006, and 12035001 is also acknowledged.

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# On the $\ell$-DLIPs of codes over finite commutative rings

Sanjit Bhowmick<EMAIL_ADDRESS>Alexandre Fotue Tabue
<EMAIL_ADDRESS>Joydeb Pal<EMAIL_ADDRESS>Department of
Mathematics, National Institute of Technology Durgapur, West Bengal, India.
Department of Mathematics, HTTC Bertoua, University of Bertoua, Bertoua,
Cameroon.  Department of Mathematics, School of Applied Sciences, Kalinga
Institute of Industrial Technology (KIIT), Deemed to be University, Odisha,
India.

###### Abstract

Generalizing the linear complementary duals, the linear complementary pairs
and the hull of codes, we introduce the concept of $\ell$-dimension linear
intersection pairs ($\ell$-DLIPs) of codes over a finite commutative ring
$(R)$, for some positive integer $\ell$. In this paper, we study $\ell$-DLIP
of codes over $R$ in a very general setting by a uniform method. Besides, we
provide a necessary and sufficient condition for the existence of a non-free
(or free) $\ell$-DLIP of codes over a finite commutative Frobenius ring. In
addition, we obtain a generator set of the intersection of two constacyclic
codes over a finite chain ring, which helps us to get an important
characterization of $\ell$-DLIP of constacyclic codes. Finally, the
$\ell$-DLIP of constacyclic codes over a finite chain ring are used to
construct new entanglement-assisted quantum error correcting (EAQEC) codes.

###### keywords:

Finite Frobenius rings, dimension of linear codes, constacyclic codes,
$\ell$-LIPs of codes, EAQEC codes. _AMS Subject Classification 2010:_ 51E22;
94B05.

††journal: submit to Discrete Mathematics

##  1 Introduction

$\ell$-linear intersection pairs ($\ell$-LIPs) of codes over finite fields
have been studied due to their wide applications in cryptography. A pair of
linear codes over finite fields is called an $\ell$-LIP if their intersection
has dimension $\ell$. This idea was introduced by Guenda et al. [7]. The
authors in [7] extended the concept of the linear complementary dual (LCD)
codes and the linear complementary pair (LCP) of codes over finite fields
(details of LCD codes, LCP of codes and their applications can be found in [1,
3, 4, 14]). In the same paper, Guenda et al. have shown that good $\ell$-LIP
of codes over finite fields exist and provided an application of linear
$\ell$-intersection pair of codes by constructing entanglement-assisted
quantum error correcting (EAQEC) codes.

In [11], Liu and Hu defined a pair of codes to be an $\ell$-LIP of codes over
finite chain rings if $\ell$ is the rank of their intersection. The notions of
the rank of linear codes and the dimension of linear codes over finite fields
coincide. Dougherty and Liu defined the rank of linear codes over finite rings
in [5]. But the rank of linear codes over finite rings is not a tool that
allows us to determine the cardinality of these linear codes. These inspiring
works highlighted the research gap in generalizing the dimension for linear
codes over finite local rings.

In this paper, we introduce the dimension of linear codes over a finite local
ring with the residue field $\mathbb{F}_{q}$ and in using the Chinese
Remainder Theorem, we extend the dimension to linear codes over a finite
commutative Frobenius ring. We generalize the $\ell$-LIP of codes over finite
fields to $\ell$-dimension LIP ($\ell$-DLIP) of codes over finite local rings
and study the characterizations of $\ell$-DLIP of codes over a finite
commutative Frobenius ring.

The paper is organized as follows. In Section 2, we recall background
materials of linear codes over a finite commutative ring. In Section 3, we
define $\ell$-DLIP of codes over a finite commutative ring and elaborate the
property of $\ell$-DLIP of codes to obtain some salient characterizations for
$\ell$-DLIP of codes over finite commutative Frobenius rings. Section 4
studies $\ell$-DLIP of constacyclic codes over a finite chain ring. Finally,
we present some results on $\ell$-DLIP of codes under the Gray map and an
application of $\ell$-DLIP of constacyclic codes over finite chain rings in
constructing new entanglement-assisted quantum error correcting codes.

##  2 Linear codes over a finite commutative ring

This section uses $R$ as a finite commutative ring with multiplicative unity
$1$ ($0\neq 1$). The ring $R$ is a principal ideal ring (PIR) if a single
element generates each of its ideals. The ring $R$ is local if it has a unique
maximal ideal. In [16, Theorem 2.2], Nechaev demonstrated that the cardinality
of any finite local ring is the power of its residue field.

######  Lemma 1.

[16, Theorem 2.2] Let $R$ be a finite local ring whose residue field is
$\mathbb{F}_{q}$ and $\textgoth{m}$ be a maximal ideal of $R$. Then there is a
positive integer $t$ (so-called the nilpotency index of $\textgoth{m}$) such
that $R$ contains a strictly descending chain of the ideals

$R\supsetneq\textgoth{m}\supsetneq\textgoth{m}^{2}\supsetneq\cdots\supsetneq\textgoth{m}^{t-1}\supsetneq\textgoth{m}^{t}=\\{0\\},$

which satisfies the conditions $t\leq\omega,|R|=q^{\omega}$ and
$|\textgoth{m}|=q^{\omega-1}.$

For all $1\leq i<t$, the quotients $\textgoth{m}^{i}/\textgoth{m}^{i+1}$ are
spaces over the field $\mathbb{F}_{q}$, and
$\mu_{i}:=\dim_{\mathbb{F}_{q}}(\textgoth{m}^{i}/\textgoth{m}^{i+1})$. The
parameters $(\mu_{0},\cdots,\mu_{t-1})$ are called the _Loewy invariants_ of a
local finite ring $R$. We have

$|\textgoth{m}^{i}|=q^{\mu_{i}+\cdots+\mu_{t-1}}$

for any $i\in\\{1,\cdots,t-1\\}$. Denote the maximal ideals of $R$ as
$\textgoth{m}_{1},\cdots,\textgoth{m}_{u}$. For any $1\leq i\leq u$, the
stationary index $s_{i}$ of $\textgoth{m}_{i}$ is defined as
$s_{i}:=\min\\{k\in\mathbb{N}\;:\;\textgoth{m}_{i}^{k}=\textgoth{m}_{i}^{k+1}\\}$.
Clearly, $R_{j}:=R/\textgoth{m}_{j}^{s_{j}}$ is a finite local ring with
maximal ideal $\textgoth{m}_{j}/\textgoth{m}_{j}^{s_{j}}$. Note that $s_{i}$
is the nilpotency index of $\textgoth{m}_{j}/\textgoth{m}_{j}^{s_{j}}$. Denote
$\mathbb{F}_{q_{j}}$, the residue field of $R_{j}$. Then we have the ring
epimorphisms

$\displaystyle\begin{array}[]{cccc}\Phi_{j}:&R&\rightarrow&R_{j}\\\
&a&\mapsto&a+\textgoth{m}_{j}\end{array}$ (3)

and $\texttt{Ker}(\Phi_{j})=\textgoth{m}_{j}^{s_{j}},$ for $1\leq j\leq u$.
The ideals $\textgoth{m}_{1},\cdots,\textgoth{m}_{u}$ are coprime and
$\bigcap_{j=1}^{u}\textgoth{m}_{j}^{s_{j}}=\\{0_{R}\\}$. Using Chinese
remainder theorem (see [15, p.224]), the ring epimorphisms $\Phi_{j}$ ($1\leq
j\leq u$) induce the following ring isomorphism as follows.

$\displaystyle\begin{array}[]{cccc}\Phi:&R&\rightarrow&R_{1}\times\cdots\times
R_{u}\\\ &a&\mapsto&\left(\Phi_{1}(a),\cdots,\Phi_{u}(a)\right).\end{array}$
(6)

The inverse of the map (6) is denoted as CRT, and $R$ is called the _Chinese
product of rings_ $\\{R_{j}\\}_{j=1}^{u}.$

Recall that the Jacobson radical of $R$, denoted as $\texttt{J}(R)$, is the
intersection of all maximal ideals of $R$. Also, the socle of $R$, denoted as
$\texttt{Soc}(R)$, is the sum of the minimal $R$-submodules of $R$. The ring
$R$ is Frobenius if the $R$-module $R$ is injective. Alternatively, $R$ is
Frobenius if $R/\texttt{J}(R)\simeq\texttt{soc}(R)$ (as $R$-modules). Any
finite PIR (so-called Chinese product of finite chain rings) is a Frobenius
ring.

######  Example 2.1.

Let $\Re_{k}:=\mathbb{F}_{2}[u_{1},u_{2},\cdots,u_{k}]$ be a finite
commutative ring such that

$\\{u_{1},u_{2},\cdots,u_{k}\\}\cap\mathbb{F}_{2}=\emptyset\text{ and
}u_{1}^{2}=u_{2}^{2}=\cdots=u_{k}^{2}=0,$

where $k$ is a positive integer. For $\emptyset\neq
A\subseteq\\{1,2,\cdots,k\\}$, we set $u_{A}:=\prod\limits_{i\in A}u_{i}$ and
$u_{\emptyset}:=1$. The ring $\Re$ is an $\mathbb{F}_{2}$-vector space with
basis $\left\\{u_{A}\;:\;A\subseteq\\{1,2,\cdots,k\\}\right\\}$. The ideals of
$\Re_{k}$ are subspaces of $\Re_{k}$ whose basis are subsets of
$\left\\{u_{A}:\;\emptyset\neq A\subseteq\\{1,2,\cdots,k\\}\right\\}$. Thus,
$\texttt{J}(\Re_{k})=\langle u_{1},u_{2},\cdots,u_{k}\rangle$ and
$\texttt{Soc}(\Re_{k})=\left\langle\prod\limits_{i=1}^{k}u_{i}\right\rangle$.
The Loewy invariants of $\Re_{k}$ is $(\mu_{0},\mu_{1},\cdots,\mu_{k})$, where
$\mu_{i}=\dim_{\mathbb{F}_{2}}\left(\texttt{J}(\Re_{k})^{i}/\texttt{J}(\Re_{k})^{i+1}\right)=\left(\begin{array}[]{c}k\\\
i\end{array}\right)$ and
$\Re_{k}/\texttt{J}(\Re_{k})\simeq\texttt{Soc}(\Re_{k})$ (as
$\Re_{k}$-modules). Therefore, $\Re_{k}$ is a finite commutative local non-
chain Frobenius ring with $\omega=2^{k}$.

######  Lemma 2.

[15, p.224] Any commutative Frobenius ring can be written as the Chinese
product of commutative local Frobenius rings.

We use this decomposition of rings to understand codes defined over finite
commutative Frobenius rings. Denote the zero vector in $R^{n}$ as 0. A linear
code $C$ of length $n$ over a finite ring $R$ is an $R$-submodule of $R^{n}.$
For any linear code $C$ over $R$ of length $n$, the minimal generating sets
for $C$ have the same cardinality, called the rank of $C$ and denoted as
$\texttt{Rk}_{R}(C)$. Thus,

$\texttt{Rk}_{R}(C):=\texttt{min}\left\\{i\in\mathbb{N}:\;\text{there exists a
monomorphism $C\hookrightarrow R^{i}$ as $R$-modules}\right\\}.$

Let $\\{\textbf{v}_{1},\textbf{v}_{2},\cdots,\textbf{v}_{s}\\}$ be a set of
non-zero vectors in $R^{n}$. Denote
$\langle\,\textbf{v}_{1},\textbf{v}_{2},\cdots,\textbf{v}_{s}\,\rangle$ as the
set of all linear combinations of vectors
$\textbf{v}_{1},\textbf{v}_{2},\cdots,\textbf{v}_{s}$. In [5], Dougherty and
Liu studied several types of independence of vectors. Recall that the set
$\\{\textbf{v}_{1},\textbf{v}_{2},\cdots,\textbf{v}_{s}\\}$ is

  * 1.

_$R$ -modular independent_ if for any $(\alpha_{1},\cdots,\alpha_{s})$ in
$R^{s}$, $\sum\limits_{i=1}^{s}\alpha_{i}\textbf{v}_{i}=\textbf{0}$ implies
$\alpha_{i}\in\textgoth{m}$, for any $1\leq i\leq s$.

  * 2.

_$R$ -independent_ if for some $(\alpha_{1},\cdots,\alpha_{s})$ in $R^{s}$,
$\sum\limits_{i=1}^{s}\alpha_{i}\textbf{v}_{i}=\textbf{0}$ implies
$\alpha_{i}\textbf{v}_{i}=\textbf{0}$, for some $1\leq i\leq s$.

The $R$-independence implies the $R$-modular independence when $R$ is local.
Thus, a generator matrix for a linear code $C$ is a matrix whose rows form a
basis for $C$. The authors in [5, Theorem 4.7] proved that those bases exist
for any code over a principal ideal ring and that the cardinality of any basis
is equal to the rank $\texttt{Rk}_{R}(C)$.

######  Example 2.2.

We consider the ring cited in Example 2.1. Let $s$ be a nonnegative integer
such that $s\leq k$ and $\\{i_{1},\cdots,i_{s}\\}$ be a subset of
$\\{1,2,\cdots,k\\}$. The set of vectors
$\\{u_{i_{1}},u_{i_{2}},\cdots,u_{i_{s}}\\}$ are $\Re_{k}$-modular
independent, but they are not $\Re_{k}$-independent. Thus, the ideal $\langle
u_{i_{1}},u_{i_{2}},\cdots,u_{i_{s}}\rangle$ of $\Re_{k}$ is a linear code
over $\Re_{k}$ of length $1$ with generator matrix
$\left(\begin{array}[]{c}u_{i_{1}}\\\ \vdots\\\ u_{i_{s}}\end{array}\right).$
Moreover, $|\langle
u_{i_{1}},u_{i_{2}},\cdots,u_{i_{s}}\rangle|=2^{\left(2^{k}-2^{k-s}\right)}$.

Let $C_{j}$ be an $n$-length code over $R_{j}$. We extend the map $\Phi$
coordinatewise to $R^{n}$ as

$\displaystyle\begin{array}[]{cccc}\Phi:&R^{n}&\rightarrow&(R_{1})^{n}\times\cdots\times(R_{u})^{n}\\\
&\textbf{a}&\mapsto&\left(\Phi_{1}(\textbf{a}),\cdots,\Phi_{u}(\textbf{a})\right),\end{array}$
(9)

where

$\displaystyle\begin{array}[]{cccc}\Phi_{j}:&R^{n}&\rightarrow&(R_{j})^{n}\\\
&(a_{1},a_{2},\cdots,a_{n})&\mapsto&\left(\Phi_{j}(a_{1}),\Phi_{j}(a_{2}),\cdots,\Phi_{j}(a_{n})\right).\end{array}$
(12)

$C$ is a linear code over $R$ if and only if for all $1\leq j\leq u$,
$\Phi_{j}(C)$ is a linear code over $R_{j}$. Since $\Phi$ is bijective,
$C=\texttt{CRT}(C_{1},C_{2},\cdots,C_{u}):=\Phi^{-1}(C_{1}\times
C_{2}\times\cdots\times C_{u}),$ where $\Phi_{j}(C)=C_{j}$ for $1\leq j\leq
u.$ In this case, $C$ is called the Chinese product of codes
$\\{C_{j}\\}_{j=1}^{u}.$

######  Lemma 3.

[5, Theorem 2.4] Let $C$ be the Chinese product of linear codes
$\\{C_{j}\\}_{j=1}^{n},$ where $C_{j}$ is a linear code over $R_{j}$, and
$\mathbb{F}_{q_{j}}$ is the residue field of $R_{j}$. Then

  1. 1.

$|C|=\prod_{i=1}^{u}|C_{j}|;$

  2. 2.

$\texttt{Rk}_{R}(C)=\texttt{max}\left\\{\texttt{Rk}_{R_{j}}(C_{j}):\;1\leq
j\leq u\right\\};$

  3. 3.

$C$ is a free code if and only if each $C_{j}$ is free with the same rank
$\texttt{Rk}_{R}(C).$

We Denote $\dim(C_{j}):=\log_{q_{j}}(|C_{j}|)$ and
$\texttt{Rank}_{q_{j}}(\mathrm{A}_{j}):=\log_{q_{j}}(|C_{j}|),$ where
$\mathrm{A}_{j}$ is a matrix whose rows span $C_{j}$. By setting
$Q:=\prod\limits_{j=1}^{u}q_{j}$ from Lemma 3, it follows that

$\displaystyle\log_{Q}(|C|)=\sum\limits_{j=1}^{u}\left(\frac{\ln(q_{j})}{\ln(Q)}\right)\dim(C_{j}).$
(13)

The positive integer $\log_{Q}(|C|)$ is called the dimension of $C$, denoted
as $\dim(C)$. Now, we consider the standard inner product to the space $R^{n}$
as follows.

$\displaystyle[\textbf{v}\,,\,\textbf{w}]=\sum_{j=1}^{n}v_{j}w_{j},$ (14)

where $\textbf{v}:=(v_{1},v_{2},\cdots,v_{n})$ and
$\textbf{w}:=(w_{1},w_{2},\cdots,w_{n})$ are vectors in $R^{n}.$ For a code
$C,$ its dual code is defined as $C^{\perp}=\\{\textbf{u}\in
R^{n}:\;[\textbf{u},\textbf{c}]=0_{R},\text{ for all }\;\textbf{c}\;\text{ in
}C\\}.$ A generator matrix for $C^{\perp}$ is called a parity-check matrix for
$C$. It is well known that for codes over Frobenius rings,
$|C|\times|C^{\perp}|=|R|^{n}$ (see [18] for the proof). Thus, for any linear
code $C$ over $R$, we have

$\displaystyle\dim(C)+\dim(C^{\perp})=n\omega.$ (15)

######  Example 2.3.

We consider the ring cited in Example 2.1. Let $s$ be a nonnegative integer
such that $s\leq k$ and $\\{i_{1},\cdots,i_{s}\\}$ be a subset of
$\\{1,2,\cdots,k\\}$. Then in $\Re_{k}$, we have

$\langle
u_{i_{1}},u_{i_{2}},\cdots,u_{i_{s}}\rangle^{\perp}=\texttt{Ann}(\langle
u_{i_{1}},u_{i_{2}},\cdots,u_{i_{s}}\rangle)=\left\langle\prod\limits_{j=1}^{s}u_{i_{j}}\right\rangle.$

Moreover,
$\left|\left\langle\prod\limits_{j=1}^{s}u_{i_{j}}\right\rangle\right|=2^{\left(2^{k-s}\right)}$.

##  3 Characterization of $\ell$-DLIP of codes over a finite local Frobenius
ring

In this section, $R$ is a finite commutative local Frobenius ring with residue
field $\mathbb{F}_{q}$, where $q$ is a prime power and $\omega$ is a positive
integer such that $|R|=q^{\omega}$.

######  Definition 3.1.

For a nonnegative integer $\ell$, a pair $\\{C,D\\}$ of linear codes of length
$n$ over $R$ is called an $\ell$-dimension linear intersection pair
($\ell$-DLIP) if $\dim(C\cap D)=\ell.$

Let $\Re$ be a finite Frobenius ring such
$\Re:=\texttt{CRT}(R_{1},R_{2},\cdots,R_{s})$. From Eq. 13, we extend the
definition of the dimension of linear codes over finite local Frobenius rings
to finite Frobenius rings. Therefore, the definition of $\ell$-DLIP of codes
over $\Re$ is given as follows: a pair $\\{C,D\\}$ of linear codes of length
$n$ over a finite Frobenius ring $\Re$ is an $\ell$-DLIP if $\dim(C\cap
D)=\ell$. Using CRT, we have the following result.

######  Proposition 1.

Let $\Re$ be a finite Frobenius ring such that
$\Re:=\texttt{CRT}(R_{1},R_{2},\cdots,R_{s})$, and $\\{C,D\\}$ be a pair of
linear codes over $R$ such that $C:=\texttt{CRT}(C_{1},C_{2},\cdots,C_{s})$
and $D:=\texttt{CRT}(D_{1},D_{2},\cdots,D_{s})$. If for each $1\leq i\leq s$,
$\\{C_{i},D_{i}\\}$ is an $\ell$-DLIP of codes over $R_{i}$, then $\\{C,D\\}$
is an $\ell$-DLIP of codes over $\Re$.

From the above definition, the following facts are immediate.

  1. Fact 1:

a linear code $C$ is LCD if $\\{C,C^{\perp}\\}$ is $0$-DLIP (for details
information, see [2]);

  2. Fact 2:

a free linear $0$-DLIP $\\{C,D\\}$ with $\dim(C)+\dim(D)=n$ is an LCP (for
details information, see [10]);

  3. Fact 3:

the dimension of hull of any linear code $C$ is $\ell$ if and only if
$\\{C,C^{\perp}\\}$ is $\ell$-DLIP (for details information about the the hull
of cyclic codes over a finite chain ring, see [17]).

Therefore, the generalization of LCD codes, LCP of codes, and the hull of
codes can be viewed as $\ell$-DLIP codes. Now, we derive the properties of
$\ell$-DLIP of codes over $R$ in terms of their generator and parity-check
matrices. In this context, the following propositions are required.

######  Proposition 2.

Let $C$ and $D$ be linear codes over $R$ of length $n$. Then

$\dim(C+D)=\dim(C)+\dim(D)-\dim(C\cap D).$

###### Proof.

Consider the map

$\begin{array}[]{cccc}\phi:&C\times D&\mapsto&C+D\\\
&(\textbf{x},\textbf{y})&\mapsto&\textbf{x}+\textbf{y}.\end{array}$

This map is an $R$-module homomorphism. By the definition of $C+D$, the map
$\phi$ is surjective. Therefore, according to the First Isomorphism Theorem,
$C\times D/\texttt{Ker}(\phi)\simeq C+D$ (as $R$-modules). Since $C\cap
D\simeq\texttt{Ker}(\phi)$ (as $R$-modules), it follows that $\frac{|C\times
D|}{|C\cap D|}=|C+D|$. Thus, $\dim(C)+\dim(D)-\dim(C\cap D)=\dim(C+D)$. ∎

######  Proposition 3.

Let $C$ be a linear code over $R$ of length $n$ and $\mathrm{A}$ be a matrix
over $R$ with $n$ column(s). Then

$\dim(\\{\textbf{x}\in
C:\;A\textbf{x}^{\top}=\textbf{0}\\})=\dim(C)-\texttt{Rank}_{q}(\mathrm{A}).$

###### Proof.

We define a map

$\begin{array}[]{cccc}\psi:&C&\rightarrow&\texttt{Im}(\psi)\\\
&\textbf{x}&\mapsto&\mathrm{A}\textbf{x}^{\top},\end{array}$

which is an $R$-module epimorphism. Since $\texttt{ker}(\psi)=\\{\textbf{x}\in
C\;:\;\mathrm{A}\textbf{x}^{\top}=\textbf{0}\\}$, it is easy to verify that
$\texttt{ker}(\psi)$ and the solution space of
$\mathrm{A}\textbf{x}^{\top}=\textbf{0}$ both are $R$-module isomorphic. By
the First Isomorphism Theorem, $C/\texttt{ker}(\psi)\simeq\texttt{Im}(\psi)$
(as $R$-modules). Thus, $\dfrac{\mid
C\mid}{\mid\texttt{ker}(\phi)\mid}=|\texttt{Im}(\psi)|$. Therefore,

$\log_{q}(\mid
C\mid)-\log_{q}(\mid\texttt{ker}(\psi)\mid)=\log_{q}(|\texttt{Im}(\psi)|).$

From the definition of the dimension of a linear code, we have
$\dim(C)-\dim(\texttt{ker}(\psi))=\dim(\texttt{Im}(\psi))$. ∎

######  Proposition 4.

Let $\\{C,D\\}$ be a pair of linear codes of length $n$ over $R$. Then

$\displaystyle\dim(C^{\perp}\cap D^{\perp})=n\omega-\dim(C)-\dim(D)+\dim(C\cap
D).$ (16)

###### Proof.

It is well known that $C^{\perp}+D^{\perp}=(C\cap D)^{\perp}.$ From
Proposition 2, we obtain

$\dim(C^{\perp}+D^{\perp})=\dim(C^{\perp})+\dim(D^{\perp})-\dim(C^{\perp}\cap
D^{\perp}).$

Besides, Eq.15 gives $\dim((C\cap D)^{\perp})=n\omega-\dim(C\cap D).$ This
means $\dim(C^{\perp}\cap D^{\perp})=n\omega-\dim(C)-\dim(D)+\dim(C\cap D)$. ∎

To establish the necessary and sufficient condition of $\ell$-DLIP of codes,
the following lemmas are needed.

######  Lemma 4.

Let $C_{i}$ be a linear code of length $n$ over $R$ with generator matrix
$\mathrm{G}_{i}$, for $i=1,2$. Then

$\texttt{Rank}_{q}\left({\begin{array}[]{ccc}\mathrm{G}_{1}\\\
\mathrm{G}_{2}\\\
\end{array}}\right)=\texttt{Rank}_{q}(\mathrm{G}_{1})+\texttt{Rank}_{q}(\mathrm{G}_{2})-\dim(C_{1}\cap
C_{2}).$

###### Proof.

We set $\dim(C_{1}\cap C_{2})=\ell.$ We know that $x\in C_{1}^{\perp}$ if and
only if $\mathrm{G}_{1}x^{\top}={\bf 0}$ and $x\in C_{2}^{\perp}$ if and only
if $\mathrm{G}_{2}x^{\top}={\bf 0}$. Thus, $x\in C_{1}^{\perp}\cap
C_{2}^{\perp}$ if and only if $\mathrm{B}x^{\top}={\bf 0},$ where
$\mathrm{B}=\left({\begin{array}[]{ccc}\mathrm{G}_{1}\\\ \mathrm{G}_{2}\\\
\end{array}}\right)$. From Proposition 3, we obtain $\dim\\{\textbf{x}\in
R^{n}\;:\;\mathrm{B}\textbf{x}^{\top}=\textbf{0}\\}=n\omega-\texttt{Rank}_{q}(\mathrm{B}).$
Combining it with Proposition 4, we get
$n\omega-\dim(C_{1})-\dim(C_{2})+\ell=n\omega-\texttt{Rank}_{q}(B),$ which
means
$\texttt{Rank}_{q}(\mathrm{B})=\texttt{Rank}_{q}(\mathrm{G}_{1})+\texttt{Rank}_{q}(\mathrm{G}_{2})-\ell$,
as $\dim(C_{i})=\texttt{Rank}_{q}(\mathrm{G}_{i})$. Therefore,

$\texttt{Rank}_{q}\left({\begin{array}[]{ccc}\mathrm{G}_{1}\\\
\mathrm{G}_{2}\\\
\end{array}}\right)=\texttt{Rank}_{q}(\mathrm{G}_{1})+\texttt{Rank}_{q}(\mathrm{G}_{2})-\dim(C_{1}\cap
C_{2}).$

∎

Combining Definition 3.1 with Lemma 4, the two results are immediate
consequences.

######  Remark 1.

Let $C_{i}$ be a linear code of length $n$ over $R$ with generator matrix
$\mathrm{G}_{i}$, for $i=1,2$. Then $\\{C_{1},C_{2}\\}$ is an $\ell$-DLIP if
and only if

$\texttt{Rank}_{q}\left({\begin{array}[]{ccc}\mathrm{G}_{1}\\\
\mathrm{G}_{2}\\\
\end{array}}\right)=\texttt{Rank}_{q}(\mathrm{G}_{1})+\texttt{Rank}_{q}(\mathrm{G}_{2})-\ell.$

######  Remark 2.

Let $C_{i}$ be a linear code of length $n$ over $R$ with parity check matrix
$\mathrm{H}_{i}$, for $i=1,2$. Then

$\dim(C_{1}^{\perp}\cap
C_{2}^{\perp})=\texttt{Rank}_{q}(\mathrm{H}_{1})+\texttt{Rank}_{q}(\mathrm{H}_{2})-\texttt{Rank}_{q}\left({\begin{array}[]{ccc}\mathrm{H}_{1}\\\
\mathrm{H}_{2}\\\ \end{array}}\right).$

######  Lemma 5.

Let $C_{i}$ be a linear code of length $n$ over $R$ with generator matrix
$\mathrm{G}_{i}$ and parity check matrix $\mathrm{H}_{i}$, for $i=1,2$. Then

$\dim(C_{1}\cap
C_{2})=\dim(C_{1})-\texttt{Rank}_{q}(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})$

or

$\dim(C_{1}\cap
C_{2})=\dim(C_{2})-\texttt{Rank}_{q}(\mathrm{H}_{1}\mathrm{G}_{2}^{\top}).$

###### Proof.

We set $d_{i}:=\texttt{Rk}_{R}(C_{i}).$ Then there is a submodule
$\textgoth{M}$ of $R^{d_{1}}$ such that

$\begin{array}[]{cccc}\phi:&\textgoth{M}&\rightarrow&C_{1}\\\
&\textbf{m}&\mapsto&\textbf{m}\mathrm{G}_{1}\end{array}$

is an $R$-module isomorphism. $x\in C_{1}\cap C_{2}$ if and only if there is
$(\alpha,\beta)$ in $\textgoth{M}\times R^{d_{2}}$ such that
$x=\alpha\mathrm{G}_{1}=\beta\mathrm{G}_{2}.$ Therefore,
$\mathrm{H}_{2}x^{\top}=\mathrm{H}_{2}(\alpha\mathrm{G}_{1})^{\top}=(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})\alpha^{\top}$
and
$\mathrm{H}_{2}x^{\top}=\mathrm{H}_{2}(\beta\mathrm{G}_{2})^{\top}=(\mathrm{H}_{2}\mathrm{G}_{2}^{\top})\beta^{\top}=\textbf{0}$,
since $\mathrm{H}_{2}\mathrm{G}_{2}^{\top}=\textbf{O}$. Thus, $C_{1}\cap
C_{2}=\\{\alpha\mathrm{G}_{1}:\;(\exists\alpha\in\textgoth{M})((\mathrm{H}_{2}\mathrm{G}_{1}^{\top})\alpha^{\top}={\bf
0})\\}$. Since $\phi$ is an isomorphism, $C_{1}\cap C_{2}$ and

$\\{\alpha\in\textgoth{M}\;:\;(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})\alpha^{\top}={\bf
0})\\}$

are isomorphic. Therefore, from Proposition 3 we have

$\dim(C_{1}\cap
C_{2})=\dim(\\{\alpha\in\textgoth{M}:\;(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})\alpha^{\top}={\bf
0})\\})=\dim(C_{1})-\texttt{Rank}_{q}(\mathrm{H}_{2}\mathrm{G}_{1}^{\top}).$

∎

Next, we give an essential characterization of $\ell$-DLIP of codes which is a
straightforward application of Lemma 5.

######  Theorem 1.

Let $C_{i}$ be a linear code of length $n$ over $R$ with generator matrix
$\mathrm{G}_{i}$ and parity check matrix $\mathrm{H}_{i}$, for $i=1,2$. Then
$\\{C_{1},C_{2}\\}$ is an $\ell$-DLIP if and only if
$\texttt{Rank}_{q}(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})=\texttt{Rank}_{q}(\mathrm{G}_{1})-\ell$
or
$\texttt{Rank}_{q}(\mathrm{H}_{1}\mathrm{G}_{2}^{\top})=\texttt{Rank}_{q}(\mathrm{G}_{2})-\ell.$

######  Example 3.1.

In $\Re_{3}:=\mathbb{F}_{2}[u_{1},u_{2},u_{3}]$, consider $C_{1}:=\langle
u_{2},u_{3}\rangle$ and $C_{2}:=\langle u_{1},u_{3}\rangle$. Then

$\mathrm{G}_{1}:=\left(\begin{array}[]{c}u_{2}\\\
u_{3}\end{array}\right),\mathrm{H}_{1}:=\left(u_{2}u_{3}\right),\,\mathrm{G}_{2}:=\left(\begin{array}[]{c}u_{1}\\\
u_{3}\end{array}\right),\text{ and }\mathrm{H}_{2}:=\left(u_{1}u_{3}\right).$

Thus, $\dim(C_{1})=\dim(C_{2})=2^{3}-2=6$. Now,
$\mathrm{H}_{2}\mathrm{G}_{1}^{\top}=\mathrm{H}_{1}\mathrm{G}_{2}^{\top}=\left(\begin{array}[]{cc}u_{1}u_{2}u_{3}&0\end{array}\right).$
Since $\langle u_{1}u_{2}u_{3}\rangle=\texttt{J}(\Re_{3})^{\perp}$, then
$\texttt{Rank}_{q}(\mathrm{H}_{1}\mathrm{G}_{2}^{\top})=\texttt{Rank}_{q}(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})=\log_{2}|\langle
u_{1}u_{2}u_{3}\rangle|=1$, and so $\dim(C_{1}\cap C_{2})=6-1=5$. Indeed,
$\\{u_{2},u_{3},u_{1}u_{2},u_{1}u_{3},u_{2}u_{3},u_{1}u_{2}u_{3}\\}\text{ and
}\\{u_{1},u_{3},u_{1}u_{2},u_{1}u_{3},u_{2}u_{3},u_{1}u_{2}u_{3}\\}$ are
$\mathbb{F}_{2}$-bases of $C_{1}$ and $C_{2}$, respectively. It deduces
$\\{u_{3},u_{1}u_{2},u_{1}u_{3},u_{2}u_{3},u_{1}u_{2}u_{3}\\}$ is an
$\mathbb{F}_{2}$-basis of $C_{1}\cap C_{2}$ and
$\left(\begin{array}[]{c}u_{3}\\\ u_{1}u_{2}\end{array}\right)$ is a generator
matrix for $C_{1}\cap C_{2}$. Hence $\\{C_{1},C_{2}\\}$ is $5$-DLIP.

######  Corollary 1.

Let $C_{i}$ be a linear code of length $n$ over $R$ with generator matrix
$\mathrm{G}_{i}$ and parity check matrix $\mathrm{H}_{i}$, for $i=1,2$. Then a
pair $\\{C_{1},C_{2}\\}$ of codes with $C_{1}+C_{2}=R^{n}$ is $\ell$-DLIP if
and only if
$\texttt{Rank}_{q}(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})=\texttt{Rank}_{q}(\mathrm{G}_{1})-\ell=n\omega-\texttt{Rank}_{q}(\mathrm{G}_{2})$
or
$\texttt{Rank}_{q}(\mathrm{H}_{1}\mathrm{G}_{2}^{\top})=\texttt{Rank}_{q}(\mathrm{G}_{2})-\ell=n\omega-\texttt{Rank}_{q}(\mathrm{G}_{1})$.

###### Proof.

Since $C_{1}+C_{2}=R^{n}$, it follows that $\dim(C_{1}+C_{2})=n\omega$. From
Proposition 2, we obtain
$\texttt{Rank}_{q}(\mathrm{G}_{1})+\texttt{Rank}_{q}(\mathrm{G}_{2})=n\omega+\ell$.
Combining it with Lemma 5 follows the proof of the theorem. ∎

In the case when $C_{2}=C_{1}^{\perp}$, we have
$\texttt{Hull}(C_{1})=\texttt{Hull}(C_{1}^{\perp})=C_{1}\cap C_{1}^{\perp}$.
Thus, we have the following result.

######  Corollary 2.

Let $C$ be a linear code over $R$ of length $n$ with generator matrix
$\mathrm{G}$ and parity check matrix $\mathrm{H}$. Then
$\dim(\texttt{Hull}(C))=\ell$ if and only if
$\texttt{Rank}_{q}(\mathrm{G}\mathrm{G}^{\top})=\texttt{Rank}_{q}(\mathrm{G})-\ell$
or
$\texttt{Rank}_{q}(\mathrm{H}\mathrm{H}^{\top})=\texttt{Rank}_{q}(\mathrm{H})-\ell.$

Now, we recall the definition of the projective module as follows: an
$R$-module $C$ is projective if there is an $R$-module $M$ such that $C\oplus
M$ is a free $R$-module.

######  Remark 3.

Let $\mathrm{B_{1}}$ and $\mathrm{B_{2}}$ be $R$-modules. If
$\mathrm{B_{1}}\oplus\mathrm{B_{2}}$ is free, then $\mathrm{B_{1}}$ and
$\mathrm{B_{2}}$ are projective.

######  Lemma 6.

[12, Theorem 2] Over chain rings, any projective module is free.

######  Theorem 2.

Let $C_{i}$ be a non-free linear code of length $n$ over $R$, for $i=1,2$ such
that $C_{1}+C_{2}=R^{n}$. Then there always exists a non-zero $\ell$-DLIP of
codes.

###### Proof.

Let $C_{1}\cap C_{2}=\\{\,\textbf{0}\,\\}$. Then we have $C_{1}\oplus
C_{2}=R^{n}$, as $C_{1}+C_{2}=R^{n}$. Since $C_{1}\oplus C_{2}$ is free, then
Remark 3 implies that $C_{1}$ as well as $C_{2}$ are projective. Now, Lemma 6
claims that $C_{1}$ as well as $C_{2}$ must be free, which is a contradiction
to the fact that $C_{1}$ and $C_{2}$ are non-free. Therefore, $C_{1}\cap
C_{2}\neq\\{\,\textbf{0}\,\\}$. ∎

######  Corollary 3.

Let $\\{C_{1},C_{2}\\}$ be a pair of linear codes over $R$ of length $n$ such
that $C_{1}+C_{2}=R^{n}$. If $\\{C_{1},C_{2}\\}$ is $0$-DLIP, then $C_{1}$ and
$C_{2}$ are free.

By setting $\ell=0$, combining Lemma 4 and Corollary 3, we have the following
result.

######  Corollary 4.

Let $C_{i}$ be a free linear code of length $n$ over $R$ with generator matrix
$\mathrm{G}_{i}$ and parity check matrix $\mathrm{H}_{i},$ for $i=1,2$, such
that $C_{1}+C_{2}=R^{n}$. Then the following statements are equivalent.

  1. 1.

$C_{1}$ and $C_{2}$ form LCP;

  2. 2.

$\left({\begin{array}[]{ccc}\mathrm{G}_{1}\\\ \mathrm{G}_{2}\\\
\end{array}}\right)$ is invertible over $R$;

  3. 3.

$\left({\begin{array}[]{ccc}\mathrm{H}_{1}\\\ \mathrm{H}_{2}\\\
\end{array}}\right)$ is invertible over $R$;

  4. 4.

$\texttt{Rank}_{q}(\mathrm{H}_{2}\mathrm{G}_{1}^{\top})=\texttt{Rank}_{q}(\mathrm{G}_{1})$
or
$\texttt{Rank}_{q}(\mathrm{H}_{1}\mathrm{G}_{2}^{\top})=\texttt{Rank}_{q}(\mathrm{G}_{2})$.

##  4 Constacyclic $\ell$-DLIP of codes over a finite chain ring

In this section, $R$ is assumed to be a finite chain ring (FCR) of invariants
$(q,t)$. Therefore, $R$ is a local ring with maximal ideal $\textgoth{m}$.
Indeed, $\textgoth{m}$ is a principal ideal. Thus,
$\textgoth{m}=\langle\gamma\rangle,$ $R/\textgoth{m}\simeq\mathbb{F}_{q}$ and
$\\{0\\}=\langle\gamma^{t}\rangle\subsetneq\langle\gamma^{t-1}\rangle\subsetneq\cdots\subsetneq\gamma\subsetneq\langle\gamma^{0}\rangle=R$,
where $t$ is called the nilpotency index of $\gamma$ and $\mathbb{F}_{q}$ is
the residue field of $R$.

We introduce the mapping

$\begin{array}[]{cccc}\pi:&R&\rightarrow&\mathbb{F}_{q}\\\
&r&\mapsto&r+\textgoth{m},\end{array}$

which is a natural epimorphism from $R$ onto $\mathbb{F}_{q}.$ This mapping
$\pi$ can be extended naturally to an epimorphism from $R^{n}$ to
$\mathbb{F}_{q}^{n}$. For a given unit $\lambda$ in $R$, a linear code $C$ of
length $n$ over $R$ is $\lambda$-constacyclic if $(\lambda
c_{n-1},c_{0},c_{1},\cdots,c_{n-2})\in C$ whenever
$(c_{0},c_{1},\cdots,c_{n-1})\in C.$ The $1_{R}$-constacyclic codes are called
cyclic codes and $(-1_{R})$-constacyclic codes are called negacyclic codes. It
is well known that the image of a $\lambda$-constacyclic code of length $n$
over $R$ under the $R$-isomorphism

$\displaystyle\begin{array}[]{cccc}\Psi:&R^{n}&\rightarrow&R[X]/\langle
X^{n}-\lambda\rangle\\\
&(c_{0},c_{1},\cdots,c_{n-1})&\mapsto&c_{0}+c_{1}x+\cdots+c_{n-1}x^{n-1},\end{array}$

is an ideal of the residue class ring $\frac{R[x]}{\langle
x^{n}-\lambda\rangle}.$ Now, for each $a(x):=\sum\limits_{i=0}^{s}a_{i}x^{i}$
with $a_{0}$ invertible in $R$, the reciprocal of $a(x)$ is denoted by
$a^{*}(x)$ and is defined by
$a^{*}(x)=x^{s}a_{0}^{-1}\sum\limits_{i=0}^{s}a_{i}x^{-i}$. If
$a(x)=a^{*}(x)$, then $a(x)$ is called self-reciprocal. Now, we focus on
$\lambda$-constacyclic $\ell$-DLIP of codes over $R$. Our next result on non-
free $\lambda$-constacyclic over $R$ follows from [8, Theorem 3.14].

######  Lemma 7.

Let $C$ be a $\lambda$-constacyclic code of length $n$ over $R$. Then there
exists a unique $(t+1)$-tuple $(F_{0},\dots,F_{t})$ of pairwise coprime
polynomials in $R[x]$ satisfying $F_{0}\cdots F_{t}=x^{n}-\lambda$, such that

$\Psi(C)=\left\langle\left\\{\gamma^{t}\widehat{\,F_{j+1}}\;:\;0\leq
j<t\right\\}\right\rangle\text{~{}and~{}}\Psi(C^{\perp})=\left\langle\left\\{\widehat{\,F_{0}\;}^{*}\right\\}\cup\left\\{\gamma^{j}\widehat{\,F_{t-j+1}}^{*}\;:\;1\leq
j\leq t-1\right\\}\right\rangle,$

where $\widehat{\,F_{i}\;}=\frac{x^{n}-\lambda}{F_{i}}$, for $0\leq i\leq t$.
Moreover,
$|C|=q^{\sum\limits_{j=0}^{t-1}{(t-j)}\deg(F_{j+1})}~{}\mbox{and}~{}|C^{\perp}|=q^{\sum\limits_{j=1}^{t}{j}\deg(F_{j+1})}$
with $F_{t+1}=F_{0}$.

Our next result about the intersection of two $\lambda$-constacyclic codes
over $R$ follows from Lemma 7, the division algorithm theorem and the notion
of least common multiple.

######  Theorem 3.

Let $C_{i}$ be a $\lambda$-constacyclic code over $R$ such that

$\Psi(C_{i})=\langle\\{\gamma^{j}\widehat{\,F_{i,j}\;}\;:\;0\leq
j<t\\}\rangle,$

where ${F}_{i,0}{F}_{i,1}\cdots{F}_{i,t}=x^{n}-\lambda$, for $i=1,2$. Then

$\displaystyle\Psi(C_{1}\cap
C_{2})=\left\langle\left\\{\gamma^{j}\texttt{lcm}(\widehat{\,F_{1,j}\;},\widehat{\,F_{2,j}\;})\;:\;0\leq
j<t\right\\}\right\rangle.$ (17)

Moreover, $\dim(C_{1}\cap
C_{2})=\sum\limits_{j=0}^{t-1}(t-j)(n-\deg(\texttt{lcm}(\widehat{\,F_{1,j}\;},\widehat{\,F_{2,j}\;}))).$

For better understanding, we discuss an example over the ring $\mathbb{Z}_{4}$
below.

######  Example 4.1.

We know that $x^{7}+1=(x+1)(x^{3}-2x^{2}+x+1)(x^{3}+x^{2}+2x+1)$ over
$\mathbb{Z}_{4}$. Let $f(x)=x^{3}+x^{2}+2x+1$, $g(x)=x^{3}-2x^{2}+x+1$ and
$h(x)=x+1$ which are basic irreducible polynomials. Let $C_{1}$ and $C_{2}$ be
two negacyclic codes of length $7$ over $\mathbb{Z}_{4}$. Notice that

$\Psi(C_{1})=\langle f(x)g(x),2f(x)\rangle\text{ and }\Psi(C_{2})=\langle
h(x)f(x),2h(x)\rangle$

are also negacyclic codes of length $7$ over $\mathbb{Z}_{4}$. Therefore,

$\displaystyle\Psi(C_{1}\cap C_{2})$ $\displaystyle=$
$\displaystyle\langle\texttt{lcm}(f(x)g(x),h(x)f(x)),2\texttt{lcm}(f(x),h(x))\rangle;$
$\displaystyle=$ $\displaystyle\langle 2(x+1)(x^{3}+x^{2}+2x+1)\rangle,$

Therefore, $\\{C_{1},C_{2}\\}$ is a $3$-DLIP of codes.

######  Corollary 5.

Let $C_{i}$ be a free $\lambda$-constacyclic code over $R$ such that
$\Psi(C_{i})=\langle F_{i}\rangle$, where $F_{i}$ divides $x^{n}-\lambda$, for
$i=1,2$. Then $C_{1}+C_{2}=R^{n}$ if and only if
$\texttt{lcm}(F_{1},F_{2})=F_{1}F_{2}.$ Moreover, $C_{1}$ and $C_{2}$ form an
LCP if and only if $\deg(F_{1}F_{2})=n$.

###### Proof.

We have $\dim(C_{i})=t(n-\deg(F_{i}))$. From Proposition 2, we get

$\dim(C_{1}+C_{2})=\dim(C_{1})+\dim(C_{2})-\dim(C_{1}\cap C_{2}).$

Since $C_{1}+C_{2}=R^{n}$, it follows that $\dim(C_{1}+C_{2})=tn$. Thus,
$\dim(C_{1}\cap C_{2})=t(n-\deg(F_{1})-\deg(F_{2}))$. By Theorem 3,
$\dim(C_{1}\cap C_{2})=t(n-\deg(\texttt{lcm}(F_{1},F_{2})))$. Hence
$\deg(\texttt{lcm}(F_{1},F_{2}))=\deg(F_{1})+\deg(F_{2}).$ ∎

Let $C_{1}$ be a $\lambda_{1}$-constacyclic code and $C_{2}$ be a
$\lambda_{2}$-constacyclic code, where
$\pi{(\lambda_{1})}\neq\pi{(\lambda_{2})}$. With this assumption, we propose
our next result.

######  Proposition 5.

Let $C_{i}$ be a free $\lambda_{i}$-constacyclic code over $R$ of length $n$,
for $i=1,2$, with $\pi{(\lambda_{1})}\neq\pi{(\lambda_{2})}$ such that
$C_{1}\cap C_{2}$ is both $\lambda_{1}$ and $\lambda_{2}$-constacyclic. Then
$C_{1}\cap C_{2}=R^{n}$ or $C_{1}\cap C_{2}=\\{\,\textbf{0}\,\\}$.

###### Proof.

Assume that $C_{1}\cap C_{2}\neq\\{\,\textbf{0}\,\\}$. Since the codes $C_{1}$
and $C_{2}$ are free, $C_{1}\cap C_{2}$ is also free. Thus, there exists
$\textbf{x}:=(x_{1},x_{2},\cdots,x_{n})$ in $C_{1}\cap C_{2}$ such that
$x_{1}$ is an invertible element in $R$. Denote $\tau_{i}(\textbf{x})$ as the
$\lambda_{i}$-constashift of x. Then
$\tau_{i}(\textbf{x})=(\lambda_{i}x_{n},x_{1},\cdots,x_{n-1})$. Since
$C_{1}\cap C_{2}$ is both $\lambda_{1}$ and $\lambda_{2}$-constacyclic, it
follows that
$\tau_{1}(\textbf{x})-\tau_{2}(\textbf{x})=((\lambda_{1}-\lambda_{2})x_{n},0,\cdots,0)\in
C_{1}\cap C_{2}$. Since $\pi{(\lambda_{1})}\neq\pi{(\lambda_{2})}$, it deduces
that $\lambda_{1}-\lambda_{2}$ is an invertible element in $R$, and so
$(1,0,\cdots,0)\in C_{1}\cap C_{2}$. Hence $C_{1}\cap C_{2}=R^{n}$. ∎

######  Remark 4.

Let $\lambda_{1}$ and $\lambda_{2}$ be two invertible elements in $R$ such
that $\pi{(\lambda_{1})}\neq\pi{(\lambda_{2})}$. Let $C_{i}$ be a free
$\lambda_{i}$-constacyclic code over $R$ of length $n$, for $i=1,2,$ such that
$C_{1}\cap C_{2}$ is both $\lambda_{1}$ and $\lambda_{2}$-constacyclic and
$C_{1}+C_{2}=R^{n}$. Then $\\{C_{1},C_{2}\\}$ is an LCP.

If we set $C_{1}=C$ and $C_{2}=C^{\perp}$, then $C$ and $C^{\perp}$ form a
free $\lambda$-constacyclic and free $\lambda^{-1}$-constacyclic codes over
$R$, respectively. We have the following result in this direction using
Corollary 5 and Remark 4.

######  Corollary 6.

Let $C$ be a free $\lambda$-constacyclic code over $R$ such that
$\Psi(C)=\langle F\rangle$ where $F$ divides $x^{n}-\lambda$, with
$\pi(\lambda^{2})=1$. Then $C$ forms an LCD code if and only if $F(x)$ is
self-reciprocal.

In order to calculate all $\lambda$-constacyclic LCD codes, we need to
consider the case $\pi{(\lambda)}\neq\pi{(\lambda)}^{-1}$. In this regard, we
state the following result from Proposition 5.

######  Corollary 7.

Let $C$ be a free $\lambda$-constacyclic code over $R$ of length $n$ such that
$\Psi(C)=\langle F\rangle$, where $F$ divides $x^{n}-\lambda$ and
$\pi{(\lambda)}\neq\pi{(\lambda)}^{-1}$. Then $C$ forms an LCD over $R$.

In the sequel, we consider the ring $\mathbb{F}_{q}[\gamma]$ with
$\gamma\neq\gamma^{2}=0$. Also, assume that $q=p^{s}$, where $p$ is a prime
number, $s$ is a positive integer, and $q\equiv 1\pmod{4}$. Thus,
$\mathbb{F}_{q}[\gamma]$ is a finite commutative chain ring and
$\texttt{J}(\mathbb{F}_{q}[\gamma])=\gamma\mathbb{F}_{q}$. Note that
$\mathbb{F}_{q}[\gamma]$ is a vector space over $\mathbb{F}_{q}$ with ordered
basis $(1,\gamma)$. In [9, Section 5] the Gray map
$\phi:\mathbb{F}_{q}[\gamma]\rightarrow\mathbb{F}_{q}^{2}$ has been defined as
below.

$\begin{array}[]{cccc}\phi:&\mathbb{F}_{q}[\gamma]&\rightarrow&\mathbb{F}_{q}^{2}\\\
&a+\gamma b&\mapsto&(\alpha b,a+b),\end{array}$

where $\alpha^{2}=-1$, for some $\alpha$ in $\mathbb{F}_{q}[\gamma]$. This
mapping is an isomorphism of vector spaces over $\mathbb{F}_{q},$ which can be
extended to $(\mathbb{F}_{q}[\gamma])^{n}\rightarrow(\mathbb{F}_{q})^{2n}$ in
the following way.

$\begin{array}[]{cccc}\phi:&(\mathbb{F}_{q}[\gamma])^{n}&\rightarrow&(\mathbb{F}_{q})^{2n}\\\
&(x_{1},x_{2},\cdots,x_{n})&\mapsto&(\phi(x_{1}),\phi(x_{2}),\cdots,\phi(x_{n})).\end{array}$

The map $\phi:(\mathbb{F}_{q}[\gamma])^{n}\rightarrow(\mathbb{F}_{q})^{2n}$ is
also an isomorphism of vector spaces over $\mathbb{F}_{q}$. Therefore, for any
pair $\\{C_{1},C_{2}\\}$ of linear codes over $\mathbb{F}_{q}[\gamma]$ of
length $n$, we have

$\dim(C_{1}\cap C_{2})=\dim(\phi(C_{1})\cap\phi(C_{2})).$

This means that $\\{C_{1},C_{2}\\}$ is an $\ell$-DLIP if and only if
$\\{\phi(C_{1}),\phi(C_{2})\\}$ is an $\ell$-LIP. Now, we define a Gray weight
function over $\mathbb{F}_{q}[\gamma]$ as

$\texttt{wt}_{G}(a+\gamma b)=\left\\{\begin{array}[]{ll}0,&\hbox{if
$a=b=0$;}\\\ 2,&\hbox{if $b\neq 0$ and $a\neq-b$;}\\\
1,&\hbox{otherwise.}\end{array}\right.$

The Gray weight over $\mathbb{F}_{q}[\gamma]$ can also be extended to
$(\mathbb{F}_{q}[\gamma])^{n}$ as follows:

$\texttt{wt}_{G}(x_{1},x_{2},\cdots,x_{n}):=\sum\limits_{i=1}^{n}\texttt{wt}_{G}(x_{i}),$

for any $(x_{1},x_{2},\cdots,x_{n})$ in $(\mathbb{F}_{q}[\gamma])^{n}$. The
Gray map $\phi:(\mathbb{F}_{q}[\gamma])^{n}\rightarrow(\mathbb{F}_{q})^{2n}$
is an $\mathbb{F}_{q}$-linear map such that
$\texttt{wt}_{H}(\phi(\textbf{x}))=\texttt{wt}_{G}(\textbf{x}),$ for any x in
$(\mathbb{F}_{q}[\gamma])^{n}$, where $\texttt{wt}_{H}$ is the Hamming weight
on $(\mathbb{F}_{q})^{2n}$. Therefore, the Gray image of any linear code $C$
over $\mathbb{F}_{q}[\gamma]$ of length $n$, is a linear code $\phi(C)$ over
$\mathbb{F}_{q}$ of length $2n$ with $\dim(C)=\dim(\phi(C))$ and
$\texttt{wt}_{H}(\phi(C))=\texttt{wt}_{G}(C)$.

Entanglement-assisted quantum error correcting (EAQEC) codes were introduced
firstly by Hsieh et al. [6]. The authors in [6] have framed EAQEC codes from
arbitrary classical linear codes. An $\left[\left[n,k,d;c\right]\right]_{q}$
EAQEC code over $\mathbb{F}_{q}$ encodes $k$ logical qudits into $n$ physical
qudits with the help of $c$ copies of maximally entangled states. We recall
the proposition [7, Proposition 4.2], which demonstrates that classical linear
codes can be used to create EAQEC codes.

######  Proposition 6.

Let $\\{C_{1},C_{2}\\}$ be an $\ell$-LIP of codes with parameters
$[n,k_{1},d_{1}]_{q}$ and $[n,k_{2},d_{2}]_{q}$, respectively. Then there
exists an $[[n,k_{2}-\ell,\min\\{d_{1}^{\perp},d_{2}\\};k_{1}-\ell]]_{q}$
EAQEC code with $d_{1}^{\perp}=d(C_{1}^{\perp})$.

Using the Gray map, Proposition 6 and Theorem 3, we have the following
corollary.

######  Corollary 8.

Let $C_{i}$ be a $\lambda$-constacyclic code over $\mathbb{F}_{q}[\gamma]$
with minimum Gray weight $\texttt{w}_{i}$ such that
$\Psi(C_{i})=\langle\\{\widehat{\,F_{i,1}\;},\,\gamma\widehat{\,F_{i,1}\;}\\}\rangle,$
where ${F}_{i,0}{F}_{i,1}{F}_{i,2}=x^{n}-\lambda$ (for $i\in\\{1,2\\}$). Then
there exists an EAQEC code with parameter

$[[2n,\tau_{2},\min\\{\texttt{w}_{1}^{\perp},\texttt{w}_{2}\\};\tau_{1}]]_{q},$

where
$\tau_{i}=2\left(\deg(F_{i,1})+\deg(\texttt{lcm}(\widehat{\,F_{1,0}\;},\widehat{\,F_{2,0}\;}))\right)+\left(\deg(F_{i,2})+\deg(\texttt{lcm}(\widehat{\,F_{1,1}\;},\widehat{\,F_{2,1}\;}))\right)-3n,$
and $\texttt{w}_{1}^{\perp}=\texttt{wt}_{G}(C_{1}^{\perp}).$

Acknowledgements The authors would like to thank the anonymous referees for
their careful reading, insightful comments, and suggestions which help us to
improve our manuscript drastically.

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