# Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two

Sanja Rukavina  Faculty of Mathematics, University of Rijeka, 51000 Rijeka,
Croatia  Vladimir D. Tonchev  Department of Mathematical Sciences, Michigan
Technological University, Houghton, MI 49931, USA

###### Abstract

The parameters 2-$(36,15,6)$ are the smallest parameters of symmetric designs
for which a complete classification up to isomorphism is yet unknown. In [5]
Bouyukliev, Fack and Winne classified all 2-$(36,15,6)$ designs that admit an
automorphism of odd prime order, and gave a partial classification of such
designs that admit an automorphism of order 2. In this paper, we give the
classification of all symmetric 2-$(36,15,6)$ designs that admit an
automorphism of order two. It is shown that there are exactly $1547701$
nonisomorphic such designs, $135779$ of which are self-dual designs. The
ternary linear codes spanned by the incidence matrices of these designs are
computed. Among these codes, there are near-extremal self-dual codes with
previously unknown weight distributions.

Mathematical subject classification (2020): 05B05, 94B05

##  1 Introduction

We assume familiarity with the basic facts and notions from the theory of
combinatorial designs [2, 7, 14, 21].

Symmetric designs with parameters $2$-$(36,15,6)$ can be obtained in many
different ways, and it is not easy to determine the first occurrence of such a
design. For example, in the first edition of Hall’s Combinatorial Theory [10],
designs with parameters $2$-$(36,15,6)$ were listed as unkonwn, while in his
1964 work [15], Mann wrote about the connection between balanced incomplete
block designs and abelian difference sets, and the existence of abelian
difference sets with parameters $(36,15,6)$. These designs have attracted much
attention, many connections have been made to other structures, and many such
designs are presently known (see, e.g., [5, 6, 20]. A main open problem is the
complete enumeration of such designs up to isomorphism. The problem of
classifying all $2$-$(36,15,6)$ designs was approached by assuming an action
of an automorphism group, leading to a partial classification. Bouyukliev,
Fack and Winne [5] gave a complete classification of all $2$-$(36,15,6)$
designs with an automorphism of odd prime order $p$, and a partial
classification for $p=2$. The partial classification from [5] did not cover
automorphisms of order two without fixed points, or involutions with more than
$10$ fixed points. A classification of symmetric $2$-$(36,15,6)$ designs with
an automorphism of order six was given in [9], and all designs on which a
cyclic automorphism group of order 4 acts standardly were constructed in [8].
In [17], $2$-$(36,15,6)$ designs with an automorphism of order two were used
to investigate the existence of extremal self-dual ternary codes of length
$36$. However, to the best of our knowledge, the enumeration of all symmetric
$2$-$(36,15,6)$ designs with an automorphism of order two has been an open
problem. Many examples of 2-$(36,15,6)$ designs having a trivial full
automorphism group are known (see, for example, [20], [22]), but a complete
classification of such designs is not known.
The links between Hadamard matrices of order 36, symmetric $2$-$(36,15,6)$
designs, and extremal ternary self-dual codes of length 36 were studied in
([17, 22]. In particular, it was proved in [17] that, up to isomorphism, there
is exactly one $2$-$(36,15,6)$ design with an an automorphism of order two
whose incidence matrix spans an extremal ternary self-dual code of length
$36$.

The main goal of this paper is to give a detailed complete classification of
symmetric $2$-$(36,15,6)$ designs with an automorphism of order two (Theorem
2.1). The ternary linear codes spanned by the incidence matrices of such
designs are computed. Among these codes, there are near-extremal self-dual
codes with previously unknown weight distributions.

The paper is structured as follows. In the next section we consider all
possible actions of an automorphism of order two on a symmetric
$2$-$(36,15,6)$, and give in detail a classification of all such symmetric
designs. In Section 3, we use the obtained designs to prove the existence of
near-extremal self-dual ternary codes of length $36$ with $104$ or $120$
codewords of minimum weight.

##  2 Classification of symmetric $2$-$(36,15,6)$ designs with an automorphism
of order two

An automorphism of a symmetric design ${\mathcal{D}}$ is a permutation on the
set of points of ${\mathcal{D}}$ that sends blocks to blocks. The action of an
automorphism of ${\mathcal{D}}$ generates the same number of point and block
orbits (see [14, Theorem 3.3]). The set of all automorphisms of
${\mathcal{D}}$ forms its full automorphism group, denoted by
$Aut({\mathcal{D}})$. A symmetric design is self-dual if it is isomorphic to
its dual design111The dual design $D^{T}$ of a design $D$ with an incidence
matrix $A$ is the design with the incidence matrix $A^{T}$..

A partial classification of symmetric $2$-$(36,15,6)$ designs with an
automorphism of order two can be found in [5], and some information for the
cases not treated there are given in [17]. In both cases, the method for
constructing orbit matrices with presumed action of an automorphism group,
which are then indexed to construct designs was used (see, for example, [9,
13]). We complete the classification using the same method. This work can be
seen as a continuation of our research started in [17]. Here we start with the
orbit matrices considered in [17], which are available at

     https://www.math.uniri.hr/~sanjar/structures/

In [17] we were interested in a construction of extremal ternary self-dual
codes of length $36$. Because of the large number of constructed designs, it
was faster to check the codes corresponding to all designs than to eliminate
the isomorphic copies of the designs first (see [17, Note 2]). Moreover, the
only information given in [17] about the case where an automorphism of order
two acts on a symmetric $2$-$(36,15,6)$ design without fixed points is that
such designs do not lead to self-dual ternary codes of length $36$. The same
is true for symmetric $2$-$(36,15,6)$ designs acted upon by an automorphism of
order two with six fixed points, for which it is falsely written that they do
not exist. The number of constructed orbit matrices is therefore not given for
all cases considered. Table 1 contains complete information about the number
of orbit matrices constructed in [17].

Number of fixed points | 0 | 4 | 6 | 8 | 10 | 12 | 16
---|---|---|---|---|---|---|---
Number of orbit matrices | 119909 | 12991 | 2150 | 670 | 56 | 311 | 83
Table 1: The number of orbit matrices for symmetric $2$-$(36,15,6)$ designs
with an automorphism of order two

In this paper we give details on the complete classification of all symmetric
$2$-$(36,15,6)$ designs with an automorphism of order two, correcting some
errors in the literature. In addition to our own computer programs, we use the
System for Computational Discrete Algebra GAP [23] and its package DESIGN [19]
to determine the automorphism groups and representatives of the isomorphism
classes of the constructed designs.

An automorphism of order two acting on $2$-$(36,15,6)$ design can act without
fixed points or have $f$ fixed points, where $f\in\\{4,6,8,10,12,14,16,18\\}$.
To our knowledge, a classification of symmetric $2$-$(36,15,6)$ designs for
the cases $f\in\\{0,12,14,16,18\\}$ cannot be found in the literature. In [5]
it is emphasized that for an action of an automorphism of order two, the most
time-consuming part is the generation of the fixed parts (parts corresponding
to orbits of length one) when $f$ is large. Since there are no constraints for
this part of the generation, the method used is too slow. We solved this
problem in [17] by reversing the order of the orbits and considering the rows
corresponding to orbits of length two first. This imposes additional
constraints on the construction of the fixed part in cases where all rows can
be constructed for the block orbits of length two.

Once we have determined the possible orbit lengths distributions, we should
consider the possible types for the rows of the orbit matrices corresponding
to the orbits of certain lengths, i.e. the representatives of the possible
types of block orbits related to the given orbit lenghts distribution. For the
cases where an automorphism of orbit two acts on the $2$-$(36,15,6)$ design
with $14$ or $18$ fixed points, we will take a closer look at the
corresponding row types to conclude that it is not possible to construct orbit
matrices in these cases. For all other cases, our computational results are
shown in the tables 2-8, where we give the number of mutually non-isomorphic
designs with the particular full automorphism group. We also give the number
of self-dual designs for all cases. The final isomorphism test yields the
classification shown in Table 9. The obtained results are compared with the
previously known results.

###  2.1 Symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order
two without fixed points

The possible action of an automorphism of order two on a symmetric
$2$-$(36,15,6)$ design without fixed points was not mentioned in [5]. Although
this case was not specifically investigated, the results presented in [9] show
that such designs exist.

For this action of an automorphism of order two all orbits are of length two
and all rows of the corresponding orbit matrices are of type

2 2 2 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0.

From $119909$ corresponding orbit matrices, we have constructed $13869$
mutually non-isomorphic $2$-$(36,15,6)$ designs (Table 2). Among the
constructed designs there are $1199$ self-dual designs.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
3888 | $E_{81}:(S_{4}\times Z_{2})$ | 1 | 24 | $Z_{3}\times D_{8}$ | 2
1944 | $(E_{27}:S_{4})\times Z_{3}$ | 1 |  | $D_{24}$ | 2
648 | $Z_{3}\times(E_{27}:D_{8})$ | 1 |  | $Z_{4}\times S_{3}$ | 2
432 | $((E_{4}:(E_{9}:Z_{3})):Z_{2}):Z_{2}$ | 1 | 18 | $Z_{3}\times S_{3}$ | 2
324 | $E_{9}\times(E_{9}:Z_{4})$ | 1 | 16 | $(Z_{4}\times Z_{2}):Z_{2}$ | 15
| $Z_{3}\times(E_{27}:E_{4})$ | 1 |  | $Z_{2}\times D_{8}$ | 3
240 | $S_{5}\times Z_{2}$ | 1 | 12 | $D_{12}$ | 11
216 | $(E_{4}:(E_{9}:Z_{3})):Z_{2}$ | 1 |  | $Z_{6}\times Z_{2}$ | 1
144 | $S_{4}\times S_{3}$ | 2 |  | $Z_{12}$ | 2
| $E_{9}:((Z_{4}\times Z_{2}):Z_{2})$ | 1 |  | $A_{4}$ | 2
72 | $Z_{2}\times S_{3}\times S_{3}$ | 2 | 8 | $Z_{4}\times Z_{2}$ | 28
| $Z_{3}\times((Z_{6}\times Z_{2}):Z_{2})$ | 1 |  | $D_{8}$ | 22
| $(Z_{3}\times A_{4}):Z_{2}$ | 2 |  | $E_{8}$ | 18
48 | $Z_{2}\times S_{4}$ | 2 | 6 | $Z_{6}$ | 6
36 | $Z_{6}\times S_{3}$ | 5 |  | $S_{3}$ | 1
| $S_{3}\times S_{3}$ | 1 | 4 | $E_{4}$ | 693
| $Z_{3}\times(Z_{3}:Z_{4})$ | 1 |  | $Z_{4}$ | 66
24 | $S_{4}$ | 6 | 2 | $Z_{2}$ | 12958
| $E_{4}\times S_{3}$ | 4 |  |  |
Table 2: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting without fixed points

###  2.2 Symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order
two acting with four fixed points

According to [5] there are $170648$ symmetric $2$-$(36,15,6)$ designs on which
an automorphism of order two acts with $4$ fixed points. After eliminating the
isomorphic designs constructed from $12991$ orbit matrices for such an action,
we also obtained $170648$ designs among which there are $17210$ self-dual
designs. These include the design leading to an extremal ternary self-dual
code of length $36$ equivalent to the Pless symmetry code $C(17)$ [17].
Further details of the constructed designs are given in Table 3.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
51840 | $O(5,3):Z_{2}$ | 1 | 32 | $E_{4}\times D_{8}$ | 12
12096 | $PSU(3,3):Z_{2}$ | 1 |  | $(Z_{8}:Z_{2}):Z_{2}$ | 1
3888 | $E_{81}:(S_{4}\times Z_{2})$ | 1 | 24 | $Z_{2}\times A_{4}$ | 10
1152 | $Ex^{+}_{32}:(S_{3}\times S_{3})$ | 1 |  | $S_{4}$ | 7
648 | $(S_{3}\times S_{3}\times S_{3}):Z_{3}$ | 1 |  | $E_{4}\times S_{3}$ | 6
432 | $S_{3}\times((S_{3}\times S_{3}):Z_{2})$ | 1 |  | $SL(2,3)$ | 3
| $((E_{4}:(E_{9}:Z_{3})):Z_{2}):Z_{2}$ | 1 |  | $D_{24}$ | 3
384 | $Ex^{+}_{32}:D_{12}$ | 2 |  | $(Z_{6}\times Z_{2}):Z_{2}$ | 2
360 | $A_{5}\times S_{3}$ | 2 |  | $Z_{4}\times S_{3}$ | 2
324 | $(Z_{3}:(E_{9}:Z_{3})):E_{4}$ | 1 | 18 | $Z_{3}\times S_{3}$ | 50
240 | $S_{5}\times Z_{2}$ | 1 |  | $E_{9}:Z_{2}$ | 18
216 | $(E_{9}:Q_{8}):Z_{3}$ | 1 |  | $D_{18}$ | 1
162 | $(E_{9}:Z_{3}):Z_{2}$ | 1 | 16 | $D_{16}$ | 4
144 | $S_{4}\times S_{3}$ | 2 |  | $Z_{2}\times D_{8}$ | 11
| $(Z_{3}\times SL(2,3)):Z_{2}$ | 2 |  | $(Z_{4}\times Z_{2}):Z_{2}$ | 11
| $E_{9}:((Z_{4}\times Z_{2}):Z_{2})$ | 1 |  | $Z_{4}\times E_{4}$ | 2
108 | $S_{3}\times(E_{9}:Z_{2})$ | 1 | 12 | $A_{4}$ | 22
96 | $D_{8}\times A_{4}$ | 1 |  | $D_{12}$ | 65
72 | $Z_{2}\times S_{3}\times S_{3}$ | 3 |  | $Z_{6}\times Z_{2}$ | 8
| $Z_{2}\times(E_{9}:Z_{4})$ | 2 | 8 | $Z_{4}\times Z_{2}$ | 118
64 | $D_{8}\times D_{8}$ | 2 |  | $E_{8}$ | 96
| $(E_{8}:E_{4}):Z_{2}$ | 1 |  | $D_{8}$ | 134
54 | $E_{9}:Z_{6}$ | 6 |  | $Z_{8}$ | 12
| $Z_{9}:Z_{6}$ | 2 | 6 | $S_{3}$ | 315
| $Z_{3}\times(E_{9}:Z_{2})$ | 1 |  | $Z_{6}$ | 458
48 | $Z_{2}\times S_{4}$ | 3 | 4 | $Z_{4}$ | 1208
| $(Z_{3}\times Q_{8}):Z_{2}$ | 1 |  | $E_{4}$ | 4405
| $GL(2,3)$ | 1 | 2 | $Z_{2}$ | 163599
36 | $S_{3}\times S_{3}$ | 17 |  |  |
| $Z_{2}\times(E_{9}:Z_{2})$ | 4 |  |  |
Table 3: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting with four fixed points

###  2.3 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing six points

The existence of $135139$ symmetric $2$-$(36,15,6)$ designs with an
automorphism of order two acting with six fixed points was proved in [5]. Our
construction yields the same number of mutually non-isomorphic designs shown
in Table 4. We obtain these designs from $2150$ orbit matrices constructed
from three types of rows of the orbit matrices corresponding to block orbits
of length one and four possible types of rows corresponding to block orbits of
length two. The constructed designs contain $8325$ self-dual designs.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
3888 | $E_{81}:(S_{4}\times Z_{2})$ | 1 | 36 | $S_{3}\times S_{3}$ | 11
1944 | $(E_{27}:S_{4})\times Z_{3}$ | 1 |  | $Z_{6}\times S_{3}$ | 7
648 | $Z_{3}\times(E_{27}:D_{8})$ | 1 | 24 | $Z_{3}\times D_{8}$ | 2
486 | $(Z_{3}\wr_{reg}Z_{3}):S_{3}$ | 1 |  | $S_{4}$ | 9
432 | $(((Z_{6}\times Z_{6}):Z_{3}):Z_{2}):Z_{2}$ | 1 |  | $E_{4}\times S_{3}$ | 4
324 | $(Z_{3}\times((E_{9}:Z_{3}):Z_{2})):Z_{2}$ | 1 |  | $D_{24}$ | 2
| $Z_{3}\times((Z_{3}\times(E_{9}:Z_{2})):Z_{2})$ | 1 |  | $Z_{2}\times A_{4}$ | 1
240 | $S_{5}\times Z_{2}$ | 1 | 18 | $E_{9}:Z_{2}$ | 50
216 | $((Z_{6}\times Z_{6}):Z_{3}):Z_{2}$ | 1 |  | $Z_{3}\times S_{3}$ | 9
162 | $Z_{3}\times((Z_{9}:Z_{3}):Z_{2})$ | 3 |  | $Z_{6}\times Z_{3}$ | 2
| $Z_{3}\times((E_{9}:Z_{3}):Z_{2})$ | 1 | 16 | $(Z_{4}\times Z_{2}):Z_{2}$ | 11
| $E_{9}\times(E_{9}:Z_{2})$ | 1 |  | $Z_{2}\times D_{8}$ | 3
144 | $S_{4}\times S_{3}$ | 2 | 12 | $D_{12}$ | 41
| $(Z_{2}\times(E_{9}:Z_{4})):Z_{2}$ | 1 |  | $Z_{6}\times Z_{2}$ | 45
108 | $Z_{3}\times S_{3}\times S_{3}$ | 2 | 10 | $D_{10}$ | 3
| $S_{3}\times(E_{9}:Z_{2})$ | 1 | 8 | $Z_{4}\times Z_{2}$ | 24
72 | $Z_{2}\times S_{3}\times S_{3}$ | 2 |  | $D_{8}$ | 53
| $Z_{3}\times((Z_{6}\times Z_{2}):Z_{2})$ | 1 |  | $E_{8}$ | 16
| $(Z_{3}\times A_{4}):Z_{2}$ | 2 | 6 | $Z_{6}$ | 120
54 | $Z_{3}\times(E_{9}:Z_{2})$ | 4 |  | $S_{3}$ | 290
| $E_{9}\times S_{3}$ | 4 | 4 | $E_{4}$ | 2594
48 | $Z_{2}\times S_{4}$ | 2 | 2 | $Z_{2}$ | 131807
Table 4: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting with six fixed points

###  2.4 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing eight points

The number of symmetric $2$-$(36,15,6)$ designs constructed in [5] with an
automorphism of order two fixing eight points is $126817$. Starting from $670$
orbit matrices, we obtain $130414$ mutually non-isomorphic designs for this
case. As in other cases, to eliminate isomorphic copies we have used

     BlockDesignIsomorphismClassRepresentatives();

in GAP. Among the constructed designs there are $16046$ self-dual designs.
Further information is contained in Table 5.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
51840 | $O(5,3):Z_{2}$ | 1 | 24 | $Z_{2}\times A_{4}$ | 5
1152 | $Ex^{+}_{32}:(S_{3}\times S_{3})$ | 1 |  | $E_{4}\times S_{3}$ | 2
648 | $(S_{3}\times S_{3}\times S_{3}):Z_{3}$ | 1 |  | $Z_{3}\times D_{8}$ | 1
432 | $S_{3}\times((S_{3}\times S_{3}):Z_{2})$ | 1 |  | $(Z_{6}\times Z_{2}):Z_{2}$ | 3
384 | $Ex^{+}_{32}:D_{12}$ | 2 | 20 | $Z_{5}:Z_{4}$ | 1
360 | $A_{5}\times S_{3}$ | 2 | 18 | $Z_{3}\times S_{3}$ | 18
320 | $(E_{16}:Z_{5}):Z_{4}$ | 1 | 16 | $E_{16}$ | 233
108 | $E_{27}:Z_{4}$ | 2 |  | $Z_{4}\times E_{4}$ | 2
96 | $A_{4}\times D_{8}$ | 1 |  | $Z_{4}:Z_{4}$ | 2
| $(Z_{2}\times S_{4}):Z_{2}$ | 1 |  | $D_{16}$ | 2
80 | $E_{16}:Z_{5}$ | 2 |  | $(Z_{4}\times Z_{2}):Z_{2}$ | 14
72 | $Z_{2}\times S_{3}\times S_{3}$ | 1 |  | $Z_{2}\times D_{8}$ | 7
| $Z_{2}\times(E_{9}:Z_{4})$ | 2 | 12 | $A_{4}$ | 36
64 | $(E_{8}:Z_{4}):Z_{2}$ | 5 |  | $D_{12}$ | 28
| $D_{8}\times D_{8}$ | 2 |  | $Z_{6}\times Z_{2}$ | 17
54 | $Z_{3}\times(E_{9}:Z_{2})$ | 2 |  | $Z_{3}:Z_{4}$ | 24
48 | $E_{4}\times A_{4}$ | 5 | 10 | $D_{10}$ | 3
42 | $Z_{2}\times(Z_{7}:Z_{3})$ | 1 | 8 | $D_{8}$ | 109
36 | $S_{3}\times S_{3}$ | 6 |  | $Z_{4}\times Z_{2}$ | 88
| $Z_{2}\times(E_{9}:Z_{2})$ | 2 |  | $E_{8}$ | 75
| $Z_{3}\times A_{4}$ | 2 | 6 | $Z_{6}$ | 373
32 | $E_{16}:Z_{2}$ | 17 |  | $S_{3}$ | 14
| $E_{4}\times D_{8}$ | 12 | 4 | $E_{4}$ | 6107
30 | $Z_{3}\times D_{10}$ | 2 |  | $Z_{4}$ | 947
|  |  | 2 | $Z_{2}$ | 122229
Table 5: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting with eight fixed points

###  2.5 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing ten points

Our results for the number of symmetric $2$-$(36,15,6)$ designs that admit an
action of an automorphism of order two with ten fixed points agree with those
in [5]. From $56$ orbit matrices constructed in [17], we obtained $64006$
mutually non-isomorphic designs, including $1282$ self-dual designs. Details
about the automorphism groups of the constructed designs are given in Table 6.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
108 | $Z_{3}\times S_{3}\times S_{3}$ | 2 | 12 | $D_{12}$ | 2
36 | $Z_{6}\times S_{3}$ | 2 |  | $Z_{6}\times Z_{2}$ | 44
| $S_{3}\times S_{3}$ | 2 | 6 | $Z_{6}$ | 234
18 | $Z_{3}\times S_{3}$ | 2 | 4 | $E_{4}$ | 1500
|  |  | 2 | $Z_{2}$ | 62218
Table 6: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting with ten fixed points

###  2.6 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing $12$ points

Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two fixing
$12$ points have not been classified so far. Due to the large number of
designs constructed from $311$ corresponding orbit matrices, the process of
eliminating isomorphic copies was most challenging in this case. On the other
hand, the results of [17] show that only these designs and the designs for the
case where an automorphism of order two fixes four points lead to self-dual
ternary codes of length $36$, which makes this case of particular interest.
After eliminating isomorphic copies, $981693$ mutually non-isomorphic designs
are obtained, of which $79295$ are self-dual. The results of the analysis of
the corresponding automorphism groups are shown in Table 7.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
51840 | $O(5,3):Z_{2}$ | 1 | 24 | $Z_{2}\times A_{4}$ | 4
12096 | $PSU(3,3):Z_{2}$ | 1 |  | $D_{24}$ | 3
1152 | $Ex^{+}_{32}:(S_{3}\times S_{3})$ | 1 |  | $(Z_{6}\times Z_{2}):Z_{2}$ | 1
432 | $((S_{3}\times S_{3}):Z_{2})\times S_{3}$ | 1 |  | $S_{4}$ | 2
384 | $Ex^{+}_{32}:D_{12}$ | 2 |  | $E_{4}\times S_{3}$ | 2
320 | $(E_{16}:Z_{5}):Z_{4}$ | 1 |  | $Z_{3}\times D_{8}$ | 1
144 | $(Z_{3}\times SL(2,3)):Z_{2}$ | 2 | 18 | $Z_{3}\times S_{3}$ | 7
108 | $Z_{3}\times S_{3}\times S_{3}$ | 2 |  | $Z_{6}\times Z_{3}$ | 14
96 | $D_{8}\times A_{4}$ | 1 | 16 | $D_{16}$ | 4
| $(Z_{2}\times S_{4}):Z_{2}$ | 1 |  | $(Z_{4}\times Z_{2}):Z_{2}$ | 14
80 | $E_{16}:Z_{5}$ | 2 |  | $E_{16}$ | 233
72 | $Z_{2}\times S_{3}\times S_{3}$ | 1 |  | $Z_{2}\times D_{8}$ | 8
64 | $(E_{8}:Z_{4}):Z_{2}$ | 5 |  | $Z_{4}:Z_{4}$ | 2
| $D_{8}\times D_{8}$ | 2 |  | $Z_{4}\times E_{4}$ | 2
| $(Z_{8}:E_{4}):Z_{2}$ | 1 | 12 | $Z_{12}$ | 38
54 | $E_{9}\times S_{3}$ | 2 |  | $D_{12}$ | 11
48 | $E_{4}\times A_{4}$ | 5 |  | $Z_{6}\times Z_{2}$ | 57
| $Z_{2}\times S_{4}$ | 1 |  | $A_{4}$ | 2
| $(Z_{3}\times Q_{8}):Z_{2}$ | 1 | 8 | $E_{8}$ | 80
| $GL(2,3)$ | 1 |  | $Z_{4}\times Z_{2}$ | 98
36 | $S_{3}\times S_{3}$ | 3 |  | $D_{8}$ | 136
| $Z_{2}\times(E_{9}:Z_{2})$ | 2 |  | $Z_{8}$ | 2
| $Z_{6}\times S_{3}$ | 2 | 6 | $Z_{6}$ | 1187
32 | $E_{16}:Z_{2}$ | 17 | 4 | $E_{4}$ | 8657
| $E_{4}\times D_{8}$ | 12 |  | $Z_{4}$ | 797
| $(Z_{8}:Z_{2}):Z_{2}$ | 1 | 2 | $Z_{2}$ | 970260
Table 7: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting with $12$ fixed points

###  2.7 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing $14$ points

As mentioned above, the main obstacle in constructing orbit matrices for an
automorphism of order two acting on symmetric $2$-$(36,15,6)$ designs with a
large number of fixed points is the construction of the fixed part for such
orbit matrices. Therefore, we first restrict our attention to the rows
corresponding to block orbits of length two. In this case, an orbit matrix
should have $11$ such rows, and there are only three possible types for these
rows (the first $14$ digits in each type correspond to the point orbits of
length one):

1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0

1 1 1 1 0 0 0 0 0 0 0 0 0 0 2 1 1 1 1 1 1 1 1 1 0

1 1 0 0 0 0 0 0 0 0 0 0 0 0 2 2 1 1 1 1 1 1 1 1 1

While trying to construct orbit matrices starting from fixed orbits takes too
long and fails, this approach shows that it is possible to construct at most
seven rows of an orbit matrix with row types corresponding to block orbits of
length two. Since we must have $11$ rows corresponding to block orbits of
length two, this means that an automorphism of order two cannot act on a
symmetric $2$-$(36,15,6)$ design with $14$ fixed points.

###  2.8 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing $16$ points

From $83$ orbit matrices for the action of an automorphism of order two on a
symmetric $2$-$(36,15,6)$ design with $16$ fixed points constructed in [17],
we obtain $67027$ mutually non-isomorphic designs. Among them, there are
$20467$ self-dual designs. This is the first time that such designs are
classified, and details about their automorphism groups are given in Table 8.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
51840 | $O(5,3):Z_{2}$ | 1 | 24 | $(Z_{6}\times Z_{2}):Z_{2}$ | 2
3888 | $E_{81}:(S_{4}\times Z_{2})$ | 1 |  |  |
1152 | $Ex^{+}_{32}:(S_{3}\times S_{3})$ | 1 |  | $E_{4}\times S_{3}$ | 2
648 | $(S_{3}\times S_{3}\times S_{3}):Z_{3}$ | 1 | 16 | $Z_{2}\times D_{8}$ | 7
432 | $S_{3}\times((S_{3}\times S_{3}):Z_{2})$ | 1 |  | $Z_{4}\times E_{4}$ | 2
384 | $Ex^{+}_{32}:D_{12}$ | 2 |  | $(Z_{4}\times Z_{2}):Z_{2}$ | 4
360 | $A_{5}\times S_{3}$ | 2 | 12 | $D_{12}$ | 22
96 | $D_{8}\times A_{4}$ | 1 |  | $Z_{6}\times Z_{2}$ | 18
72 | $Z_{2}\times S_{3}\times S_{3}$ | 1 | 10 | $Z_{10}$ | 5
| $Z_{2}\times(E_{9}:Z_{4})$ | 2 | 8 | $Z_{4}\times Z_{2}$ | 54
36 | $S_{3}\times S_{3}$ | 6 |  | $E_{8}$ | 67
64 | $D_{8}\times D_{8}$ | 2 |  | $D_{8}$ | 22
36 | $Z_{2}\times(E_{9}:Z_{2})$ | 2 | 6 | $Z_{6}$ | 66
32 | $E_{4}\times D_{8}$ | 12 | 4 | $E_{4}$ | 1334
24 | $Z_{2}\times A_{4}$ | 5 |  | $Z_{4}$ | 114
| $Z_{3}\times D_{8}$ | 1 | 2 | $Z_{2}$ | 65268
Table 8: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
acting with $16$ fixed points

###  2.9 Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two
fixing $18$ points

There is only one type for the rows of the orbit matrix corresponding to the
block orbits of length two for an action of an automorphism of order two that
fixes $18$ points on a symmetric $2$-$(36,15,6)$ design:

1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1

It is not difficult to see that it is not possible to combine two rows of this
type to obtain the rows corresponding to the block orbits of length two of an
orbit matrix for a symmetric $2$-$(36,15,6)$ design. Therefore, an
automorphism of order two cannot act on a symmetric $2$-$(36,15,6)$ design
with $18$ fixed points.

###  2.10 Enumeration of symmetric $2$-$(36,15,6)$ designs with an
automorphism of order two

Here we summarise the results of the analysis of all possible cases for the
action of an automorphism of order two on symmetric $2$-$(36,15,6)$ designs
and give the main result of this paper. In the previous sections, we have
eliminated isomorphic copies among the designs related to a particular action
of an automorphism of order two. To obtain a classification of all symmetric
$2$-$(36,15,6)$ designs with an automorphism of order two, we need to
eliminate isomorphic designs that admit different actions of an automorphism
of order two. If the full automorphism group of a design is of order two, then
such a design admits only one action of an automorphism order two. However, if
the full automorphism group is of composite order and contains different
conjugacy classes of a subgroup of order two, it is possible that such a
design admits different actions of an automorphism of order two (with respect
to the number of fixed points). We have thus added the numbers of designs with
the full automorphism group of order two for all possible actions considered
in the previous sections and checked the remaining designs for isomorphism.
Our analysis leads to a complete classification of symmetric $2$-$(36,15,6)$
designs admitting an automorphism of order two, which is summarised in the
following theorem.

###### Theorem 2.1

An automorphism of order two acts on a symmetric $2$-$(36,15,6)$ design with
$f$ fixed points, where $f\in\\{0,4,6,8,10,12,16\\}$. Up to isomorphism, there
are $1547701$ symmetric $2$-$(36,15,6)$ designs admitting an automorphism of
order two with the full automorphism groups as given in Table 9. Amog them
there are $135779$ self-dual designs.

The order | The structure | No. of | The order | The structure | No. of
---|---|---|---|---|---
of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs |  of $Aut\mathcal{(D)}$ |  of $Aut\mathcal{(D)}$ | designs
51840 | $O(5,3):Z_{2}$ | 1 | 48 | $E_{4}\times A_{4}$ | 5
12096 | $PSU(3,3):Z_{2}$ | 1 | 42 | $Z_{2}\times(Z_{7}:Z_{3})$ | 1
3888 | $E_{81}:(S_{4}\times Z_{2})$ | 1 | 36 | $S_{3}\times S_{3}$ | 19
1944 | $(E_{27}:S_{4})\times Z_{3}$ | 1 |  | $Z_{3}\times A_{4}$ | 2
1152 | $Ex^{+}_{32}:(S_{3}\times S_{3})$ | 1 |  | $Z_{6}\times S_{3}$ | 7
648 | $(S_{3}\times S_{3}\times S_{3}):Z_{3}$ | 1 |  | $Z_{2}\times(E_{9}:Z_{2})$ | 4
| $Z_{3}\times(E_{27}:D_{8})$ | 1 |  | $Z_{3}\times(Z_{3}:Z_{4})$ | 1
486 | $(Z_{3}\wr_{reg}Z_{3}):S_{3}$ | 1 | 32 | $E_{4}\times D_{8}$ | 12
432 | $S_{3}\times((S_{3}\times S_{3}):Z_{2})$ | 1 |  | $E_{16}:Z_{2}$ | 17
| $((E_{4}:(E_{9}:Z_{3})):Z_{2}):Z_{2}$ | 1 |  | $(Z_{8}:Z_{2}):Z_{2}$ | 1
384 | $Ex^{+}_{32}:D_{12}$ | 2 | 30 | $Z_{3}\times D_{10}$ | 2
360 | $S_{3}\times A_{5}$ | 2 | 24 | $Z_{2}\times A_{4}$ | 10
324 | $E_{9}\times(E_{9}:Z_{4})$ | 1 |  | $S_{4}$ | 13
| $(Z_{3}\times(E_{9}:Z_{3})):E_{4}$ | 1 |  | $SL(2,3)$ | 3
| $Z_{3}\times(E_{27}:E_{4})$ | 1 |  | $D_{24}$ | 5
320 | $(E_{16}:Z_{5}):Z_{4}$ | 1 |  | $Z_{3}\times D_{8}$ | 3
240 | $Z_{2}\times S_{5}$ | 1 |  | $(Z_{6}\times Z_{2}):Z_{2}$ | 3
216 | $(E_{4}:(E_{9}:Z_{3})):Z_{2}$ | 1 |  | $E_{4}\times S_{3}$ | 6
| $(E_{9}:Q_{8}):Z_{3}$ | 1 |  | $Z_{4}\times S_{3}$ | 2
162 | $(E_{27}:Z_{3}):Z_{2}$ | 1 | 20 | $Z_{5}:Z_{4}$ | 1
| $Z_{3}\times(Z_{9}:Z_{6})$ | 3 | 18 | $E_{9}:Z_{2}$ | 68
| $Z_{3}\times(E_{9}:Z_{6})$ | 1 |  | $Z_{6}\times Z_{3}$ | 16
| $E_{9}\times(E_{9}:Z_{2})$ | 1 |  | $D_{18}$ | 1
144 | $S_{3}\times S_{4}$ | 2 |  | $Z_{3}\times S_{3}$ | 88
| $(Z_{3}\times SL(2,3)):Z_{2}$ | 2 | 16 | $D_{16}$ | 4
| $E_{9}:((Z_{4}\times Z_{2}):Z_{2})$ | 1 |  | $(Z_{4}\times Z_{2}):Z_{2}$ | 25
108 | $Z_{3}\times S_{3}\times S_{3}$ | 2 |  | $E_{16}$ | 233
| $E_{27}:Z_{4}$ | 2 |  | $Z_{2}\times D_{8}$ | 11
| $(E_{9}:Z_{2})\times S_{3}$ | 1 |  | $Z_{4}:Z_{4}$ | 2
96 | $(Z_{2}\times S_{4}):Z_{2}$ | 1 |  | $Z_{4}\times E_{4}$ | 2
| $D_{8}\times A_{4}$ | 1 | 12 | $A_{4}$ | 62
80 | $E_{16}:Z_{5}$ | 2 |  | $Z_{12}$ | 40
72 | $Z_{2}\times S_{3}\times S_{3}$ | 3 |  | $Z_{6}\times Z_{2}$ | 70
| $Z_{2}\times(E_{9}:Z_{4})$ | 2 |  | $D_{12}$ | 79
| $Z_{3}\times((Z_{6}\times Z_{2}):Z_{2})$ | 1 |  | $Z_{3}:Z_{4}$ | 24
| $(Z_{3}\times A_{4}):Z_{2}$ | 2 | 10 | $D_{10}$ | 6
64 | $D_{8}\times D_{8}$ | 2 |  | $Z_{10}$ | 5
| $(E_{8}:Z_{4}):Z_{2}$ | 5 | 8 | $D_{8}$ | 235
| $(Z_{8}:E_{4}):Z_{2}$ | 1 |  | $Z_{4}\times Z_{2}$ | 180
54 | $E_{9}:Z_{6}$ | 6 |  | $E_{8}$ | 108
| $Z_{3}\times(E_{9}:Z_{2})$ | 7 |  | $Z_{8}$ | 14
| $E_{9}\times S_{3}$ | 6 | 6 | $Z_{6}$ | 2444
| $Z_{9}:Z_{6}$ | 2 |  | $S_{3}$ | 620
48 | $Z_{2}\times S_{4}$ | 3 | 4 | $Z_{4}$ | 3132
| $(Z_{3}\times Q_{8}):Z_{2}$ | 1 |  | $E_{4}$ | 11693
| $GL(2,3)$ | 1 | 2 | $Z_{2}$ | 1528339
Table 9: Symmetric $2$-$(36,15,6)$ designs with an automorphism of order two

For more detailed information on the structure of automorphism groups of
orders greater than or equal to $486$, we refer the reader to [9].

When classifying combinatorial structures using algorithms and programs, which
have been proven to provide accurate results in many previous cases, possible
errors are mainly due to the human factor and errors that occur when entering,
manipulating or analyzing a large amount of obtained data. In addition, when
working with a very large data set, hardware-related errors are also possible,
although these are very rare. Finally, the transcription of the final results
is also subject to errors.

To verify the correctness of the results presented in this paper, we applied
several techniques. First of all, we compared our results with known partial
results. Our results are in complete agreement with the results published in
[5] for the cases where an automorphism of order two acts on a symmetric
$2$-$(36,15,6)$ design with $4$, $6$ or $10$ fixed points, which gives us even
more confidence in the procedures we have performed. Moreover, they agree with
the results of [9] for the vast majority of cases where the full automorphism
group contains the subgroup that is isomorphic to the cyclic group of order
six. For the few cases where our results do not match those in [5, 9], we
double-checked our results. These include the number of mutually non-
isomorphic designs (we constructed more designs), the determinacy of the full
automorphism group structure, and the pairing designs that are not self-dual.
For the cases not previously discussed in the literature (an automorphism of
order two acts on a symmetric $2$-$(36,15,6)$ design without fixed points or
with more than $10$ points), we proceeded in the same way as for the confirmed
previously known results. We performed the isomorphism test multiple times and
with different groupings of designs, paired designs that are not self-dual,
and checked the presented data multiple times to avoid errors in the
presentation of the results.

##  3 Near-extremal ternary self-dual codes of length $36$ related to the
symmetric $2$-$(36,15,6)$ designs admitting an automorphism of order two

We assume familiarity with the basic facts and concepts from the error-
correcting codes [3, 12, 22].

The sparsity of extremal self-dual ternary codes has recently aroused interest
in near extremal ternary self-dual codes [1, 11, 18]. In this section, we
consider such codes that can be obtained from the symmetric $2$-$(36,15,6)$
designs with an automorphism of order two. In this part of our work we use the
computer algebra system MAGMA [4].

The minimum weight $d$ of a ternary self-dual code of length $n$ divisible by
12 satisfies the upper bound $d\leq n/4+3$ [12, 9.3]. A self-dual ternary code
of length $n$ divisible by 12 with a minimum weight $d$ is called extremal if
$d=n/4+3$ [12], and near-extremal if $d=n/4$ [1].

Recently, it was proved in [17] that there exists only one symmetric
$2$-$(36,15,6)$ design with an automorphism of order two whose incidence
matrix generates an extremal ternary self-dual code of length $36$. This
design admits the action of an automorphism of order two with four fixed
points and yields the only known extremal ternary self-dual code of length
$36$, namely the Pless symmetry code $C(17)$ [16].

The possible weight enumerators of near-extremal self-dual codes of length
$36$ are determined in [1] and are given by

$1+\alpha
y^{9}+(42840-9\alpha)y^{12}+(1400256+36\alpha)y^{15}+(18452280-84\alpha)y^{18}$
$+(90370368+126\alpha)y^{21}+(162663480-126\alpha)y^{24}+(97808480+84\alpha)y^{27}$
(1) $+(16210656-36\alpha)y^{30}+(471240+9\alpha)y^{33}+(888-\alpha)y^{36}.$

Moreover, the number of codewords of minimum weight in a ternary near-extremal
self-dual code of length 36 is $A_{9}=8\beta$ for some $\beta$ in the range
$1\leq\beta\leq 111$ [1, page 1830], and examples of near-extremal ternary
self-dual codes of length $36$ are known for
$A_{9}\in\\{8\beta|\beta\in\\{9,12,14,16,17,...,83,85,90,93\\}\\}$ (see [1,
11]). The question of the existence of codes for other values of $A_{9}$ is
still open. Here we consider self-dual ternary codes arising from symmetric
$2$-$(36,15,6)$ designs admitting an automorphism of order two in order to
find examples of self-dual near-extremal ternary codes of length $36$ for
values of $A_{9}$ for which the existence has not yet been proven.

As indicated in [17, Table 3], an incidence matrix of a symmetric
$2$-$(36,15,6)$ design with an automorphism of order two can generate a self-
dual ternary code of length $36$ only if the corresponding automorphism of
order two fixes four or twelve points. We consider all such designs from
Section 2.2 and Section 2.6 and the results of our analysis are summarized in
Table 10.

| designs | self-dual | inequivalent | codes with | different
---|---|---|---|---|---
|  | ternary codes | codes | $mw=9$ | $A_{9}$
$4$ fixed points | 170648 | 70774 | 2995 | 932 | 44
$12$ fixed points | $981693$ | 391720 | 2363 | 48 | 22
Table 10: Near-extremal self-dual ternary codes of length $36$ constructed
from $2$-$(36,15,6)$ designs with an automorphism of order two

The number of codewords with minimum weight in the obtained codes is given in
Table 11, where $i$ represents the number of fixed points for an action of an
automorphism of order two on a symmetric $2$-$(36,15,6)$ design.

i | $A_{9}$
---|---
4 | $112,128,136,152,160,168,176,184,192,200,208,216,224,232,240,248,256,264,$
| $272,280,288,296,304,312,320,328,336,344,352,360,368,376,384,392,400,408,$
| $416,424,432,440,448,464,480,544$
12 | $104,120,136,152,168,184,200,216,232,248,264,280,296,312,328,392,424,472,$
| $488,552,584,680$
Table 11: The number of codewords with minimum weight

Since near-extremal ternary self-dual codes with $A_{9}=8\cdot 13=104$ and
$A_{9}=8\cdot 15=120$ were not constructed in [1, 11], we have obtained first
examples of such codes. Therefore, we present incidence matrices for one
$2$-$(36,15,6)$ design that yield a self-dual near-extremal ternary code for
each of these cases in the Appendix. Details on analysis of obtained self-dual
ternary $[36,18,9]$ codes with $A_{9}=104$ or $A_{9}=120$ are given in Table
12.

| number of designs |  inequivalent $[36,18,9]$ |  the order of $Aut\mathcal{(C)}$
---|---|---|---
$A_{9}=104$ | 12 | 2 | 32, 48
$A_{9}=120$ | 12 | 7 | 24
Table 12: Self-dual ternary $[36,18,9]$ codes with $A_{9}=104$ or $A_{9}=120$

Our analysis, together with the results of [1, 11], leads to the following
conclusion.

###### Proposition 3.1

There is a ternary near-extremal self-dual code of length 36 with weight
enumerator (1) for
$A_{9}=\alpha\in\\{8\beta|\beta\in\\{9,12,...,83,85,90,93\\}\\}$.

Acknowledgement

The first author is supported by Croatian Science Foundation under the project
4571.

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##  4 Appendix

The incidence matrix of a symmetric $2$-$(36,15,6)$ design that yield a near-
extremal ternary self-dual code of length $36$ with $A_{9}=104$.

      [ 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
      [ 1 1 1 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 ]
      [ 1 1 1 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 ]
      [ 1 1 0 1 0 0 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 1 1 ]
      [ 1 1 0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 ]
      [ 1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 1 1 ]
      [ 1 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 ]
      [ 1 0 1 0 0 1 0 0 0 1 1 1 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 0 ]
      [ 1 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 0 0 1 0 1 0 1 ]
      [ 1 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 ]
      [ 1 0 0 1 1 0 0 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 1 1 1 1 0 0 1 ]
      [ 1 0 0 0 1 1 1 0 1 0 0 1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 ]
      [ 1 0 0 0 1 1 1 0 1 0 0 1 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 ]
      [ 1 0 0 0 0 0 0 1 1 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 0 ]
      [ 1 0 0 0 0 0 0 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 ]
      [ 0 1 1 1 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 ]
      [ 0 1 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 1 0 1 1 0 1 0 1 1 0 1 ]
      [ 0 1 1 0 0 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 1 1 1 0 ]
      [ 0 1 0 1 1 0 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 ]
      [ 0 1 0 1 1 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 1 1 0 0 0 1 0 0 1 1 0 0 1 0 0 ]
      [ 0 1 0 0 1 1 0 1 0 1 1 0 1 0 0 0 0 1 1 0 0 0 1 0 0 1 0 0 1 0 0 1 0 1 1 0 ]
      [ 0 1 0 0 1 1 0 1 0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 1 0 1 0 0 1 ]
      [ 0 1 0 0 0 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 0 1 0 1 1 1 1 0 0 0 0 1 1 0 0 0 ]
      [ 0 1 0 0 0 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 ]
      [ 0 0 1 1 0 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 ]
      [ 0 0 1 1 0 0 0 0 0 1 0 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 0 ]
      [ 0 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 0 0 1 0 ]
      [ 0 0 1 0 1 0 1 0 1 1 1 0 0 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 0 0 0 1 ]
      [ 0 0 1 0 1 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 1 0 1 0 1 0 0 0 0 1 0 1 0 1 0 1 ]
      [ 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 ]
      [ 0 0 0 1 0 1 1 0 0 0 1 0 1 0 0 1 0 0 0 1 1 1 0 1 0 1 0 0 0 1 1 0 0 1 1 0 ]
      [ 0 0 0 1 0 1 1 0 0 0 1 0 0 1 1 0 0 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 1 0 0 1 ]
      [ 0 0 0 1 0 1 0 1 1 1 0 1 1 0 1 0 1 0 1 0 0 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 ]
      [ 0 0 0 1 0 1 0 1 1 1 0 1 0 1 0 1 0 1 0 1 1 0 0 0 1 0 1 0 1 0 0 0 0 0 1 0 ]
      [ 0 0 0 0 1 0 1 1 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 1 1 ]
      [ 0 0 0 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 0 0 1 1 1 1 1 1 0 0 0 0 ]

The incidence matrix of a symmetric $2$-$(36,15,6)$ design that yield a near-
extremal ternary self-dual code of length $36$ with $A_{9}=120$.

        [ 1 1 1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
        [ 1 1 1 1 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 ]
        [ 1 1 1 0 1 0 0 1 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 ]
        [ 1 1 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 ]
        [ 1 1 1 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 1 ]
        [ 1 1 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 1 0 0 ]
        [ 1 0 1 0 1 1 0 0 1 1 1 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 1 1 ]
        [ 1 0 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 1 0 0 1 1 ]
        [ 1 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 1 1 0 0 1 1 0 ]
        [ 1 0 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 1 0 0 1 ]
        [ 1 0 0 0 1 0 0 1 1 0 0 0 0 0 1 0 0 1 1 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 0 0 ]
        [ 1 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 0 ]
        [ 1 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 ]
        [ 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 1 1 0 0 1 1 0 0 0 ]
        [ 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 0 ]
        [ 0 1 1 0 0 1 1 1 1 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 ]
        [ 0 1 0 1 1 0 1 1 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1 ]
        [ 0 1 0 1 1 0 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 1 0 0 1 1 1 1 0 ]
        [ 0 1 0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 1 0 1 1 0 1 0 0 0 0 1 0 1 1 0 1 1 0 1 ]
        [ 0 1 0 0 1 1 1 0 0 1 1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 1 1 0 0 0 0 ]
        [ 0 1 0 0 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 0 1 1 0 0 0 1 0 1 ]
        [ 0 1 0 0 0 1 0 1 0 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 1 0 0 1 0 1 0 ]
        [ 0 1 0 0 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 ]
        [ 0 1 0 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 0 0 0 1 0 1 0 1 1 0 1 0 0 0 1 0 ]
        [ 0 0 1 1 1 0 1 1 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 ]
        [ 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 1 1 1 1 0 0 0 0 0 0 ]
        [ 0 0 1 1 0 0 0 0 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 0 0 1 0 0 0 1 1 1 1 0 0 1 ]
        [ 0 0 1 1 0 0 0 0 1 0 0 1 0 1 0 1 1 0 0 1 1 0 0 0 1 0 0 0 1 0 1 1 0 1 1 0 ]
        [ 0 0 1 0 1 0 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 1 1 0 ]
        [ 0 0 1 0 1 0 0 0 0 0 1 1 0 1 0 1 1 0 1 0 0 1 1 1 1 0 0 1 0 0 0 0 1 0 0 1 ]
        [ 0 0 1 0 0 1 0 1 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 0 1 0 1 1 0 0 1 0 1 0 1 0 ]
        [ 0 0 1 0 0 1 0 1 0 1 0 0 0 1 1 0 1 0 1 0 1 0 0 0 0 1 1 1 0 0 0 1 0 1 0 1 ]
        [ 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 ]
        [ 0 0 0 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 1 0 ]
        [ 0 0 0 0 1 1 1 0 1 0 0 0 1 1 1 0 0 0 0 0 1 0 0 1 1 0 0 1 0 1 0 1 1 0 1 0 ]
        [ 0 0 0 0 1 1 1 0 1 0 0 0 1 1 0 1 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 0 0 1 0 1 ]