equivalent to $j_1\in\{1,\hdots,\nu n+k+1\},\hdots,j_{n-k}\in\{n-k,\hdots,(\nu+1)n\}$, $j_{n-k+1}\in\{n+1,\hdots,(\nu+1)n+k+1\},\hdots,j_{(n-k)(L+1)}\in\{(L+1)n-k,\hdots,(\nu+1+L)n\}$.

Throughout this paper, we will use the following notations: For a finite field $\mathbb F$, we set $t:=|\mathbb F|^{-1}$. Moreover, we denote a real valued function $f(t)$ in the variable $t$ by $O(t^n)$ for some $n\in\mathbb N$ if $\lim_{t\rightarrow 0}\frac{f(t)}{t^n}\leq C$ for some constant $C\in\mathbb R$. Moreover, the following theorem will be used frequently throughout this paper:

[@sch Corollary 1]\[sz\] \
(a) For $r\in\mathbb N$, consider $f\in\mathbb F[x_1,\hdots,x_r]$ with total degree $d\geq 0$. Then, $f$ has at most $d\cdot|\mathbb F|^{n-1}$ zeros.\
(b) Let $f\in\mathbb F[x_1,\hdots,x_r]$ be a nonzero polynomial of total degree $d$. Moreover, let $v_1,\hdots, v_r$ be selected at random independently and uniformly from $\mathbb F$. Then, the probability that $(v_1,\hdots, v_r)$ is a zero of $f$ is at most $d\cdot t$.

Results for $(n,n-1,1)$ convolutional codes
===========================================

As a starting point, in this section, we want to consider unit memory convolutional codes of high rate, i.e. $\delta=1$ and $k=n-1$. According to Theorem \[dual\], these codes are dual to the $(n,1,1)$ convolutional codes, which should therefore also be treated in this section. With this choice of parameters one has $L=\lfloor\frac{\delta}{k}\rfloor+\lfloor\frac{\delta}{n-k}\rfloor=1+\lfloor\frac{1}{n-1}\rfloor$. Hence, $L=1$ for $n\geq 3$ and $L=2$ for $n=2$.

$(2,1,1)$ convolutional codes
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[@diss 