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\title{\bf Nonexistence of certain singly even self-dual codes
with minimal shadow}

% input author, affilliation, address and support information as follows;
% the address should include the country, and does not have to include
% the street address 

\author{Stefka Bouyuklieva\\ %\thanks{Supported by NASA grant ABC123.}\\
\small Faculty of Mathematics and Informatics\\[-0.8ex]
\small Veliko Tarnovo University\\[-0.8ex] 
\small 5000 Veliko Tarnovo, Bulgaria.\\
\small\tt stefka@uni-vt.bg\\
\and
Masaaki Harada \qquad  Akihiro Munemasa\\
\small Research Center for Pure and Applied Mathematics\\[-0.8ex]
\small Graduate School of Information Sciences\\[-0.8ex]
\small Tohoku University\\[-0.8ex]
\small Sendai 980--8579, Japan.\\
\small\tt mharada@m.tohoku.ac.jp, munemasa@math.is.tohoku.ac.jp
}

% \date{\dateline{submission date}{acceptance date}\\
% \small Mathematics Subject Classifications: comma separated list of
% MSC codes available from http://www.ams.org/mathscinet/freeTools.html}

\date{\dateline{Jan 1, 2012}{Jan 2, 2012}\\
\small Mathematics Subject Classifications: 94B05}

\begin{document}

\maketitle

% E-JC papers must include an abstract. The abstract should consist of a
% succinct statement of background followed by a listing of the
% principal new results that are to be found in the paper. The abstract
% should be informative, clear, and as complete as possible. Phrases
% like "we investigate..." or "we study..." should be kept to a minimum
% in favor of "we prove that..."  or "we show that...".  Do not
% include equation numbers, unexpanded citations (such as "[23]"), or
% any other references to things in the paper that are not defined in
% the abstract. The abstract will be distributed without the rest of the
% paper so it must be entirely self-contained.

\begin{abstract}
It is known that there is no extremal singly even
self-dual $[n,n/2,d]$ code with minimal shadow
for $(n,d)=(24m+2,4m+4)$, $(24m+4,4m+4)$, 
$(24m+6,4m+4)$, $(24m+10,4m+4)$ and $(24m+22,4m+6)$.
In this paper, we study singly even self-dual codes with minimal shadow 
having minimum weight $d-2$ for these $(n,d)$.
For $n=24m+2$, $24m+4$ and $24m+10$, 
we show that the weight enumerator of a singly even self-dual 
$[n,n/2,4m+2]$ code with minimal shadow is uniquely determined
and we also show that there is no 
singly even self-dual $[n,n/2,4m+2]$ code with minimal shadow 
for $m \ge 155$, $m \ge 156$ and $m \ge 160$, respectively.
We demonstrate that the weight enumerator of a singly even self-dual 
code with minimal shadow is not uniquely determined
for parameters $[24m+6,12m+3,4m+2]$ and
$[24m+22,12m+11,4m+4]$.

  % keywords are optional
  \bigskip\noindent \textbf{Keywords:} 
 self-dual code, shadow, weight enumerator
 \end{abstract}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}


A (binary) code $C$ of length $n$ is a vector subspace of
$\FF_2^n$, where $\FF_2$ denotes the finite field of order $2$.
The {\em dual} code $C^{\perp}$ of $C$ is defined as
$
C^{\perp}=
\{x \in \FF_2^n \mid x \cdot y = 0 \text{ for all } y \in C\},
$
where $x \cdot y$ is the standard inner product.
A code $C$ is called
{\em self-dual} if $C = C^{\perp}$.
Self-dual codes are divided into two classes.
A self-dual code $C$ is {\em doubly even} if all
codewords of $C$ have weight divisible by four, and {\em
singly even} if there is at least one codeword of weight $\equiv 2
\pmod 4$.
Let $C$ be a singly even self-dual code and
let $C_0$ denote the
subcode of codewords having weight $\equiv0\pmod4$.
Then $C_0$ is a subcode of codimension $1$.
The {\em shadow} $S$ of $C$ is defined to be $C_0^\perp \setminus C$.
Shadows for self-dual codes were introduced by Conway and
Sloane~\cite{C-S}
in order to 
derive new upper bounds for the minimum weight of
singly even self-dual codes.
By considering shadows, Rains~\cite{Rains} showed that
the minimum weight $d$ of a self-dual code of length $n$
is bounded by
$d  \le 4 \lfloor{\frac {n}{24}} \rfloor + 6$
if $n \equiv 22 \pmod {24}$,
$d  \le 4  \lfloor{\frac {n}{24}} \rfloor + 4$
otherwise.
A self-dual code meeting the bound is called  {\em extremal}.



% , and to
% provide restrictions on the weight enumerators of %% extremal
% singly even self-dual codes.
% Let $C$ be a singly even self-dual code of 
% length $n$ and $S$ be the shadow of $C$.

% From now on, suppose that $n \not\equiv 22 \pmod {24}$.
Let $C$ be a singly even self-dual code of
length $n$ with shadow $S$.
Let $d(S)$ denote the minimum weight of $S$.
% Then we have
% %\begin{equation}\label{eq:K}
% \[
% r \le d(S) \le \frac{n}{2}+4-2d(C),
% \]
% %\end{equation}
% where $r=4,1,2$ and $3$ if $n \equiv 0,2,4$ and $6 \pmod 8$,
% respectively~\cite{BG}.
We say that $C$ is a code with {\em minimal shadow} 
if $r = d(S)$, where 
$r=4,1,2$ and $3$ if $n \equiv 0,2,4$ and $6 \pmod 8$,
respectively.
The concept of self-dual codes with minimal shadow was introduced 
in~\cite{performance}. 
In that paper, different types of self-dual codes with the same 
parameters were compared with regard to the decoding error probability. 
% In particular, it was shown that an extremal singly even
% self-dual code of length $24m+8$ with minimal shadow is
% better than any extremal doubly even self-dual code of that length.
In~\cite{BV}, the connection between singly even self-dual 
codes with minimal shadow of some lengths, combinatorial designs and 
secret sharing schemes was considered. 
It was shown in~\cite{BW} that there is no extremal singly even
self-dual code with minimal shadow
for lengths $24m+2$, $24m+4$,
$24m+6$, $24m+10$ and $24m+22$.
In~\cite{BV}, it was shown that the weight enumerator of a 
(non-extremal) singly even self-dual $[24m+2, 12m+1, 4m+2]$ code 
with minimal shadow is uniquely determined
for each positive integer $m$. These motivate us to study singly even
self-dual codes with minimal shadow having minimum weight two less than
the hypothetical extremal case.


% The main aim of this paper is to show that
% the lengths of singly even self-dual codes with minimal shadow
% having minimum weight $4m+2$ are bounded for lengths
% $24m+2$, $24m+4$ and $24m+10$.
% give an upper bound on the lengths for which
% there is a singly even self-dual code with minimal shadow
% having minimum weight $4m+2$ for lengths
% $24m+2$, $24m+4$ and $24m+10$.
% 
% for $(n,d)=(24m+2,4m+2)$, $(24m+4,4m+2)$ and $(24m+10,4m+2)$.
% $(24m+6,4m+2)$ and $(24m+22,4m+4)$.
% The powerful tool in the study of this paper is
% both theoretical and numerical approaches on the weight enumerators.

The main aim of this paper is to investigate singly even 
self-dual codes with minimal shadow having minimum weight 
$4m+2$ for the lengths $24m+2$,
$24m+4$ and $24m+10$. 
We show that the weight enumerator of a singly even self-dual 
code with minimal shadow having minimum weight $4m+2$
is uniquely determined for lengths $24m+4$ and $24m+10$.
For lengths $24m+2$, $24m+4$ and $24m+10$,
nonnegativity of the coefficients of weight enumerators shows 
that there is no such code for $m$ sufficiently large.
We also show that the uniqueness of the weight enumerator
fails for the parameters $[24m+6,12m+3,4m+2]$ and
$[24m+22,12m+11,4m+4]$.


The paper is organized as follows.
In Section~\ref{sec:2}, 
we review the results given by Rains~\cite{Rains}.
In Section~\ref{sec:24m+2}, we show
that there is no singly even self-dual $[24m+2,12m+1,4m+2]$
code with minimal shadow for $m \ge 155$.
In Sections~\ref{sec:24m+4} and~\ref{sec:24m+10}, 
for parameters $[24m+4,12m+2,4m+2]$ and $[24m+10,12m+5,4m+2]$,
we show that there is no 
singly even self-dual code with minimal shadow for $m \ge 156$
and for $m \ge 160$, respectively.
Finally, in Section~\ref{sec:rem}, we 
demonstrate that the weight enumerator of a singly even self-dual 
code with minimal shadow is not uniquely determined
for parameters $[24m+6,12m+3,4m+2]$ and
$[24m+22,12m+11,4m+4]$.


% The paper is organized as follows.
% In Section~\ref{sec:2}, 
% % some preliminaries are given.  In particular, 
% we review the results given by Rains~\cite{Rains}.
% Some basic results which are used in this paper are given.
% In Section~\ref{sec:24m+2}, we investigate weight enumerators of
% singly even self-dual $[24m+2,12m+1,4m+2]$ codes with minimal shadow.
% Nonnegativity of the coefficients of weight enumerators shows 
% that there is no singly even self-dual $[24m+2,12m+1,4m+2]$
% code with minimal shadow for $m \ge 155$.
% In Sections~\ref{sec:24m+4} and~\ref{sec:24m+10}, 
% we investigate weight enumerators of
% singly even self-dual codes with minimal shadow
% for parameters $[24m+4,12m+2,4m+2]$ and $[24m+10,12m+5,4m+2]$,
% respectively.
% For these parameters, it is shown
% that the weight enumerator of a singly even self-dual 
% code with minimal shadow is uniquely determined for each length,
% and we also show that there is no 
% singly even self-dual code with minimal shadow for $m \ge 156$
% and for $m \ge 160$, respectively.
% Finally, in Section~\ref{sec:rem}, we 
% demonstrate that the weight enumerator of a singly even self-dual 
% code with minimal shadow is not uniquely determined
% for parameters $[24m+6,12m+3,4m+2]$ and
% $[24m+22,12m+11,4m+4]$.



All computer calculations in this paper
were done with the help of 
the algebra software {\sc Magma}~\cite{Magma} and
the mathematical softwares {\sc Maple} and {\sc Mathematica}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Preliminaries}\label{sec:2}

% A (binary) $[n,k]$ {\em code} $C$ is a $k$-dimensional vector subspace
% of $\FF_2^n$.
% The parameter $n$ is called the {\em length} of $C$.
% The {\em weight} $\wt(x)$ of a vector $x \in \FF_2^n$ is
% the number of non-zero coordinates of $x$.
% The vectors of $C$ are called {\em codewords} of $C$.
% The minimum non-zero weight of all codewords in $C$ is called
% the {\em minimum weight} $d(C)$ of $C$ and an $[n,k]$ code with minimum
% weight $d$ is called an $[n,k,d]$ code.
% The \textit{dual code} $C^{\perp}$ of a code
% $C$ of length $n$ is defined as
% $
% C^{\perp}=
% \{x \in \FF_2^n \mid x \cdot y = 0 \text{ for all } y \in C\},
% $
% where $x \cdot y$ is the standard inner product.
% A code $C$ is called \textit{self-dual} if $C = C^{\perp}$.
% It is  known that a self-dual code of length $n$ exists
% if and only if  $n$ is even, and
% a doubly even self-dual code of length $n$
% exists if and only if $n$ is divisible by eight.
% A self-dual code $C$ is {\em doubly even} if all
% codewords of $C$ have weight divisible by four, and 
% {\em singly even} if there is at least one codeword of 
% weight $\equiv 2 \pmod 4$.
% %It is  known that a self-dual code of length $n$ exists
% %if and only if  $n$ is even, and
% %a doubly even self-dual code of length $n$
% %exists if and only if $n$ is divisible by eight.

Let $C$ be a singly even self-dual code of
length $n$ with shadow $S$.
% Let $A_i$ and $B_i$ $(i=0,1,\ldots,n)$
% be the numbers of vectors of
% weight $i$ in $C$ and $S$, respectively.
% The weight enumerators $W_C(y)$ and $W_S(y)$ of $C$ and $S$ are given by
% $\sum_{i=0}^n A_i y^i$
% and 
% $\sum_{i=d(S)}^{n-d(S)} B_i y^i$, respectively.
Write $n=24m+8l+2r$, where $m$ is an integer, $l\in\{0,1,2\}$ and
$r\in\{0,1,2,3\}$. 
The weight enumerators $W_C(y)$ and $W_S(y)$ of $C$ and $S$
are given by (\cite[(10), (11)]{C-S})
\begin{align}\label{eq:WC}
W_C(y) &= \sum_{i=0}^{12m+4l+r}a_i y^{2i}
=
\sum_{j=0}^{3m+l}
c_j(1+y^2)^{12m+4l+r-4j}(y^2(1-y^2)^2)^j,
\\
\label{eq:WS}
W_S(y)  &= \sum_{i=0}^{6m+2l}b_i y^{4i+r}
= \sum_{j=0}^{3m+l}(-1)^j c_j2^{12m+4l+r-6j}y^{12m+4l+r-4j}(1-y^4)^{2j},
\end{align}
respectively, for suitable integers $c_j$.
%Using this, the possible weight enumerators of
%singly even self-dual codes of length $n$ and minimum weight $d$
%can be determined for a given pair $(n,d)$.
%Usually, the possible weight enumerators are not be determined
%uniquely for a given pair $(n,d)$.
% However, the weight enumerator of an $s$-extremal singly even
% self-dual code is uniquely determined
% for a given pair $(n,d)$~\cite{BG}.
%
% Write $n=24m+8l+2r$, where $m$ is an integer, $l\in\{0,1,2\}$ and
% $r\in\{0,1,2,3\}$. 
Let
\begin{equation}\label{4a8}
(1+y^2)^{n/2-4j}(y^2(1-y^2)^2)^j=
\sum_{i=0}^{12m+4l+r} \alpha'_{i,j} y^{2i}\quad(0\leq j\leq 3m+l).
\end{equation}
Then
\begin{equation}\label{4b0}
\alpha_{i,j}'=\begin{cases}
0&\text{if $0\leq i<j\leq 3m+l$,}\\
1&\text{if $0\leq i=j\leq 3m+l$.}
\end{cases}
\end{equation}
This implies that the $(3m+l+1)\times(3m+l+1)$ matrix $[\alpha'_{i,j}]$
%is invertible, since it is lower triangular. 
is invertible, since it is unitriangular.
Let $[\alpha_{i,j}]$ be its inverse matrix. 
Then by~\eqref{4b0}, we have
\begin{equation}\label{4b1}
\alpha_{i,j}=\begin{cases}
0&\text{if $0\leq i<j\leq 3m+l$,}\\
1&\text{if $0\leq i=j\leq 3m+l$,}
\end{cases}
\end{equation}
and
\begin{equation}\label{4by}
y^{2i}=\sum_{j=0}^{3m+l}\alpha_{j,i}(1+y^2)^{n/2-4j}(y^2(1-y^2)^2)^j
\quad(0\leq i\leq 3m+l)
\end{equation}
by~\eqref{4a8}.
By~\eqref{eq:WC}, \eqref{4b1} and~\eqref{4by}, we obtain  
%Since
%\begin{align*}
%\sum_{j=0}^{3m+l}a_j y^{2j}
%&\equiv 
%A(1,y)\pmod{y^{6m+2l+1}}
%&&\text{(by (\ref{4a1}))}
%\nexteq
%\sum_{i=0}^{3m+l} c_i(1+y^2)^{n/2-4i}(y^2(1-y^2)^2)^i
%&&\text{(by (\ref{4a2}))}
%\nexteq
%\sum_{i=0}^{3m+l} c_i
%\sum_{j=0}^{12m+4l+r} \alpha'_{ji} y^{2j}
%&&\text{(by (\ref{4a8}))}
%\\&\equiv 
%\sum_{j=0}^{3m+l} \sum_{i=0}^{3m+l}
%\alpha'_{ji} c_i y^{2j}
%\pmod{y^{6m+2l+1}},
%\end{align*}
%we have
%\[a_j=\sum_{i=0}^{3m+l}\alpha'_{ji} c_i.\]
%This implies
%\begin{align}
%c_i&=\sum_{j=0}^{3m+l} \alpha_{ij} a_j 
%\notag\nexteq
%\sum_{j=0}^{i} \alpha_{ij} a_j
%&&\text{(by (\ref{4b1}))}.
%\label{4a10}
%\end{align}
\begin{equation}\label{4a10}
c_i=\sum_{j=0}^{i} \alpha_{i,j} a_j.
\end{equation}

\begin{lemma}\label{lem:alpha}
For $1\leq i\leq 3m+l$, we have
\begin{equation}\label{eq:alpha}
\alpha_{i,0}=
-\frac{n}{2i}
\sum_{\substack{0\leq t\leq n/2+1-6i\\ %t\leq i-1\\ 
t+i\text{ is odd}}}
(-1)^t\binom{\frac{n}{2}+1-6i}{t}
\binom{\frac{n-7i-t-1}{2}}{\frac{i-t-1}{2}}.
\end{equation}
\end{lemma}
\begin{proof}
For $1 \le i$, 
\begin{align*}
\alpha_{i,0}=&
-\frac{n}{2i}[\text{coeff.\ of } y^{i-1} \text{ in }
(1+y)^{-n/2-1+4i}(1-y)^{-2i}],
\end{align*}
\cite{Rains}.
Since
\begin{align*}
&(1+y)^{-n/2-1+4i}(1-y)^{-2i}\\
&=(1-y^2)^{-n/2-1+4i}(1-y)^{n/2+1-6i} \\
&=(1-y^2)^{-n/2-1+4i}
\sum_{t=0}^{n/2+1-6i}(-1)^t\binom{\frac{n}{2}+1-6i}{t}y^t,
\end{align*}
we have
\begin{align*}
\alpha_{i,0}&=
-\frac{n}{2i}
\sum_{t=0}^{n/2+1-6i}(-1)^t\binom{\frac{n}{2}+1-6i}{t}
[\text{coeff.\ of } y^{i-1} \text{ in }
(1-y^2)^{-n/2-1+4i} y^t]
\\&=
-\frac{n}{2i}
\sum_{\substack{0\leq t\leq n/2+1-6i\\ %t\leq i-1\\
t+i\text{ is odd}}}
(-1)^t\binom{\frac{n}{2}+1-6i}{t}
(-1)^{(i-t-1)/2}\binom{-\frac{n}{2}-1+4i}{\frac{i-t-1}{2}}.
%\\&=
%-\frac{n}{2i}
%\sum_{\substack{0\leq t\leq n/2+1-6i\\ t\leq i-1\\ t+i\text{ odd}}}
%(-1)^t\binom{\frac{n}{2}+1-6i}{t}
%\binom{\frac{n-7i-t-1}{2}}{\frac{i-t-1}{2}}.
\end{align*}
The result follows by applying the formula 
\[(-1)^j\binom{-n}{j}=\binom{n+j-1}{j}.\]
\end{proof}





Write
\begin{equation*}\label{4c8}
(-1)^j2^{n/2-6j}y^{n/2-4j}(1-y^4)^{2j}=
\sum_{i=0}^{6m+2l} \beta'_{i,j} y^{4i+r}
\quad(0\leq j\leq 3m+l).
\end{equation*}
Since $n/2-4j=4(3m+l-j)+r$, we have
\begin{equation*}\label{4d0}
\beta_{i,j}'=\begin{cases}
0&\text{if $i<3m+l-j$,}\\
(-1)^j 2^{n/2-6j}&\text{if $i=3m+l-j$.}
\end{cases}
\end{equation*}
This implies that the $(3m+l+1)\times(3m+l+1)$ matrix $[\beta'_{i,3m+l-j}]$
%is invertible, since it is lower triangular. Thus, the matrix
is invertible, since it is lower triangular such that
the diagonal elements are not zeros. Thus, the matrix
$[\beta'_{i,j}]$ is also invertible. 
Let $[\beta_{i,j}]$ be its inverse matrix. Then 
\begin{equation}\label{4c9}
y^{4i+r}=\sum_{j=0}^{3m+l} \beta_{j,i}(-1)^j 2^{n/2-6j}y^{n/2-4j}(1-y^4)^{2j}
\quad(0\leq i\leq 3m+l).
\end{equation}
Moreover,
$[\beta_{3m+l-i,j}]$ is the inverse of the lower triangular matrix
$[\beta'_{i,3m+l-j}]$, and so lower triangular as well, and
\[\beta_{3m+l-j,j}={\beta'}^{-1}_{j,3m+l-j}.\]
Thus
\begin{equation}\label{4d1}
\beta_{i,j}=\begin{cases}
0&\text{if $i>3m+l-j$,}\\
(-1)^{3m+l-j} 2^{6(3m+l-j)-n/2}&\text{if $i=3m+l-j$.}
\end{cases}
\end{equation}
By~\eqref{eq:WS}, \eqref{4c9} and~\eqref{4d1}, we obtain
%Since
%\begin{align*}
%\sum_{j=0}^{3m+l}b_j y^{4j+r}
%&\equiv 
%S(1,y)\pmod{y^{12m+4l+r+1}}
%&&\text{(by (\ref{4a3}))}
%\nexteq
%\sum_{i=0}^{3m+l} c_i(-1)^i2^{n/2-6i}y^{n/2-4i}(1-y^4)^{2i}
%&&\text{(by (\ref{4a4}))}
%\nexteq
%\sum_{i=0}^{3m+l} c_i
%\sum_{j=0}^{6m+2l} \beta'_{ji} y^{4j+r}
%&&\text{(by (\ref{4c8}))}
%\\&\equiv 
%\sum_{j=0}^{3m+l}\sum_{i=0}^{3m+l} 
% \beta'_{ji} c_i y^{4j+r}
%\pmod{y^{12m+4l+r+1}},
%\end{align*}
%we have
%\[b_j=\sum_{i=0}^{3m+l}\beta'_{ji} c_i.\]
%This implies
%\begin{align}
%c_i=\sum_{j=0}^{3m+l} \beta_{ij} b_j 
%\notag\nexteq
%\sum_{j=0}^{3m+l-i} \beta_{ij} b_j
%&&\text{(by (\ref{4d1}))}.
%\label{4c10}
%\end{align}
\begin{equation}\label{4c10}
c_i=\sum_{j=0}^{3m+l-i} \beta_{i,j} b_j.
\end{equation}


\begin{lemma}[Rains~\cite{Rains}]\label{lem:6b}
For $1\leq i\leq3m+l$ and $0\leq j\leq3m+l$ with $i+j\leq 3m+l$, we have
\begin{equation}\label{eq:beta}
\beta_{i,j}=(-1)^i2^{-n/2+6i}\frac{3m+l-j}{i}\binom{3m+l+i-j-1}{3m+l-i-j}.
\end{equation}
\end{lemma}


From~\eqref{4a10} and~\eqref{4c10}, we have
\begin{equation}\label{eq:ci}
c_i=\sum_{j=0}^{i} \alpha_{i,j} a_j=\sum_{j=0}^{3m+l-i} \beta_{i,j} b_j.
\end{equation}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Now let $C$ be a singly even self-dual $[24m+8l+2r,12m+4l+r,4m+2]$ code
with minimal shadow.
Suppose that $(l,r) \in \{(0,1),(0,2),(1,1)\}$.
Since the minimum weight of $C$ is $4m+2$, we have
\begin{equation}\label{eq:ai}
a_0=1,a_1=a_2=\cdots = a_{2m}=0.  
\end{equation}
Since the minimum weight of the shadow is $1$ or $2$, we have
\begin{equation}\label{eq:bi}
\begin{cases}
b_0=1                               & \text{ if } m=1, \\
b_0=1, b_1=b_2= \cdots =b_{m-1}=0   & \text{ if } m \ge 2.
\end{cases}
\end{equation}
From~\eqref{eq:ci}, \eqref{eq:ai} and~\eqref{eq:bi},
we have
\begin{equation}\label{eq:c_i-all}
c_i=
\begin{cases} 
\alpha_{i,0} & \text{ if }i=0,1,\ldots,2m, \\
\beta_{i,0}  & \text{ if }i=2m+l+1,2m+l+2,\ldots,3m+l. 
\end{cases}
\end{equation}
Suppose that $l=0$.
From~\eqref{eq:ci}, \eqref{eq:ai}, 
\eqref{eq:bi} and~\eqref{eq:c_i-all}, we obtain
\begin{align}\label{eq:1}
c_{2m}=&\alpha_{2m,0}
=\beta_{2m,0} + \beta_{2m,m}b_m,
\\
c_{2m-1} =&
\alpha_{2m-1,0}
=\beta_{2m-1,0} + \beta_{2m-1,m}b_m +\beta_{2m-1,m+1}b_{m+1}.
\label{eq:2}
\end{align}
Suppose that $l=1$.
From~\eqref{eq:ci}, \eqref{eq:ai}, 
\eqref{eq:bi} and~\eqref{eq:c_i-all}, we obtain
\begin{align}\label{eq:3}
c_{2m}=&\alpha_{2m,0}=\beta_{2m,0}+\beta_{2m,m}b_{m}+\beta_{2m,m+1}b_{m+1},
\\
c_{2m+1}
=&\alpha_{2m+1,0}+\alpha_{2m+1,2m+1}a_{2m+1}
=\beta_{2m+1,0}+\beta_{2m+1,m}b_m.
\label{eq:4}
\end{align}


%%%%%%%%%%%%%%%%
\section{Singly even
self-dual $[24m+2,12m+1,4m+2]$ codes with minimal shadow}\label{sec:24m+2}

It was shown in~\cite{BV} that
the weight enumerator of a singly even self-dual $[24m+2,12m+1,4m+2]$
code with minimal shadow is uniquely determined for each length.
In this section, we show that 
there is no singly even self-dual $[24m+2,12m+1,4m+2]$ code
with minimal shadow for $m \ge 155$.

Suppose that $m \ge 1$.
Let $C$ be a singly even self-dual $[24m+2,12m+1,4m+2]$ code
with minimal shadow.
The weight enumerators of $C$ and its shadow $S$ are written 
as in~\eqref{eq:WC} and~\eqref{eq:WS}, respectively.

% From \eqref{eq:ci}, \eqref{eq:ai} and \eqref{eq:bi},
% we have
% \begin{equation}\label{eq:c_i:24m+2}
% c_i=
% \begin{cases} 
% \alpha_{i,0} & \text{ if }i=0,1,\ldots,2m, \\
% \beta_{i,0}  & \text{ if }i=2m+1,2m+2,\ldots,3m. 
% \end{cases}
% \end{equation}
% Hence, $c_i$ $(i=0,1,\ldots,3m)$ 
% do not depend on the choice of $a_j$ $(j=0,1,\ldots,2m)$
% and $b_j$ $(j=0,1,\ldots,m-1)$.
% This means that the weight enumerator of $C$ is uniquely
% determined for each length.
% 
% 
% Hence, we have
% \begin{equation}\label{eq:1}
% \alpha_{2m,0}= \beta_{2m,0} + \beta_{2m,m}b_m.
% \end{equation}
From~\eqref{eq:alpha},
\begin{align*}
\alpha_{2m,0}=
% -\frac{24m+2}{4m}[\text{coeff.\ of } y^{2m-1} \text{ in }
%(1+y)^{-12m-1-1+8m}(1-y)^{-4m}].
% (1+y)^{-4m-2}(1-y)^{-4m}]\\
% =&
\frac{12m+1}{m}\binom{5m}{m-1}.
% =
% \frac{12m+1}{4m+1}\binom{5m}{m}.
\end{align*}
From~\eqref{4d1},
\[
\beta_{2m,m}=\frac{1}{2}.
\]
From~\eqref{eq:beta},
\begin{align*}
\beta_{2m,0} %=&2^{-12m-1+12m}\frac{3m}{2m}\binom{3m+2m-1}{3m-2m}
=\frac{3}{2^2}\binom{5m-1}{m}
=\frac{3(4m+1)}{5m}\binom{5m}{m-1}.
\end{align*}
%According to~\cite[p.~135]{Rains}, 
% Since
% \begin{align*}
%  (1+y)^{-4m-2}(1-y)^{-4m} = (1-y^2)^{-4m-2}(1-y)^{2},
% \end{align*}
% we have
% \begin{align*}
% \alpha_{2m,0}=&
% \frac{24m+2}{2m}\binom{5m}{m-1}.
% \end{align*}
From~\eqref{eq:1}, 
\begin{align*}
b_m=\frac{\alpha_{2m,0}-\beta_{2m,0}}{\beta_{2m,m}}  
%=&  \frac{24m+2}{m}\binom{5m}{m-1}
% -\frac{3}{2}\binom{5m-1}{m} \notag \\
=\frac{4(24m+1)}{5m}\binom{5m}{m-1}.
%% B gave and H checked.
\end{align*}
\begin{remark}
Unfortunately, $b_m$ was incorrectly  reported
in~\cite{ZMFG}. The correct formula for $b_m$ is given in~\cite{ZMFG2}. 
We showed that $b_m$ is always a positive integer (see~\cite{ZMFG2}).
\end{remark}

%
% bm
% \begin{proposition}
% The rational number $b_m$ in \eqref{eq:bm} is a positive integer
% for all $m\geq1$.
% \end{proposition}
% \begin{proof}
% Observe that $\frac{1}{4m+1}\binom{5m}{m}$ is 
% the generalized Catalan number $C_5(m)$ (see~\cite{Sury}),
% so $b_m=\frac{4(24m+1)}{5}C_5(m)$.
% If $m \equiv 1 \pmod 5$, then $24m+1$ is divisible by $5$. Thus
% $b_m$ is an integer.
% If $m\not\equiv1\pmod5$, then $m$ is not of the form
% $\frac{5^m-1}{4}$, where $m$ is a positive integer. 
% By~\cite[Theorem~2]{Sury} (see also~\cite[Proposition~2]{Konvalinka}),
% $C_5(m)$ is divisible by $5$, and hence $b_m$ is an integer in this
% case also.
% \end{proof}
% 
% \begin{remark}
% It was claimed in~\cite[Theorem 18, part 1]{ZMFG} that
% there is no singly even self-dual $[24m+2,12m+1,4m+2]$
% code with minimal shadow if 
% $b_m$ is not an integer (see also~\cite{ZMFG2}).
% However, the above proposition shows that
% $b_m$ is always a positive integer.
% This means that~\cite[Theorem 18, part 1]{ZMFG} is vacuous.
% \end{remark}


% Hence, we have
% \begin{equation}\label{eq:2}
% \alpha_{2m-1,0}= \beta_{2m-1,0} + \beta_{2m-1,m}b_m +\beta_{2m-1,m+1}b_{m+1}.
% \end{equation}
From~\eqref{eq:alpha},
\begin{align*}
\alpha_{2m-1,0}=&
% -\frac{24m+2}{4m-2}[\text{coeff.\ of } y^{2m-2} \text{ in }
% % (1+y)^{-12m-1-1+8m-4}(1-y)^{-4m+2}].
% (1+y)^{-4m-6}(1-y)^{-4m+2}]\\
% =&
%-\frac{24m+2}{4m-2}
-\frac{12m+1}{2m-1}
\left(
\binom{5m+4}{m-1}
+28\binom{5m+3}{m-2}
+70\binom{5m+2}{m-3}
\right.
\\ &
\left.
+28\binom{5m+1}{m-4}
+\binom{5m}{m-5}
\right)
\\ =&
-
\frac{8(12m+1)(376m^3- 4m^2  + 5m +1)}
{(4m+2)(4m+3)(4m+4)(4m+5)}
\binom{5m}{m-1}.
\end{align*}
From~\eqref{4d1},
\[
\beta_{2m-1,m+1}=-\frac{1}{2^7}.
\]
From~\eqref{eq:beta}, 
\begin{align*}
\beta_{2m-1,0}
%=&(-1)2^{-12m-1+12m-6}\frac{3m}{2m-1}\binom{3m+2m-1-1}{3m-2m+1}
%\\
=&-\frac{1}{2^{7}}\frac{3m}{2m-1}\binom{5m-2}{m+1}
=-\frac{1}{2^{4}}
\frac{3(4m-1)(4m+1)}{5(5m-1)(m+1)}\binom{5m}{m-1},\\
\beta_{2m-1,m}
%=&(-1)2^{-12m-1+12m-6}\frac{3m-m}{2m-1}\binom{3m+2m-1-m-1}{3m-2m+1-m} \\
%=&-2^{-7}\frac{2m}{2m-1}\binom{4m-2}{1}\\
=&-\frac{m}{2^{5}}.
\end{align*}
%According to~\cite[p.~135]{Rains}, 
% Since
% \begin{align*}
% (1+y)^{-4m-6}(1-y)^{-4m+2} = (1-y^2)^{-4m-6}(1-y)^{8}
% \end{align*}
% and 
% \[
% (1-y)^8=
% 1 - 8y + 28y^2 - 56y^3 + 70y^4 - 56y^5 + 28y^6 - 8y^7 + y^8,
% \]
From~\eqref{eq:2}, 
\begin{align*}
b_{m+1} =&\frac{\alpha_{2m-1,0}- \beta_{2m-1,0} - 
\beta_{2m-1,m}b_m}{\beta_{2m-1,m+1}}
% \\ =&
% \frac{\alpha_{2m-1,0}
% +2^{-7}\frac{3m}{2m-1}\binom{5m-2}{m+1}
% +2^{-7}4mb_m}{-\frac{1}{2^7}} %-2^{-7}
% \\ =&
% (-2^7)\alpha_{2m-1,0}
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}-4mb_m
% 
% \\ =&
% 2^7 \frac{24m+2}{4m-2}
% \Big(
% \binom{5m+4}{m-1}
% +28\binom{5m+3}{m-2}
% +70\binom{5m+2}{m-3}
% \\ &
% +28\binom{5m+1}{m-4}
% +\binom{5m}{m-5}
% \Big)
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}-4mb_m
% \\ =&
% 2^7 \frac{24m+2}{4m-2}
% \Big(
% \binom{5m+4}{m-1}
% +28\binom{5m+3}{m-2}
% +70\binom{5m+2}{m-3}
% +28\binom{5m+1}{m-4}
% \\ &
% +\binom{5m}{m-5}
% \Big)
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}
% -4m \Big(
% \frac{4(24m+1)}{5(4m+1)}\binom{5m}{m}
% %\frac{24m+2}{2m}\Big(2 \binom{5m}{m-1}\Big) -\frac{3}{2}\binom{5m-1}{m}
% \Big)\\
% \\ =&
% 2^7 
% \Big(
% \frac{8m(12m+1)(376m^3- 4m^2  + 5m +1)}
% {(4m+1)(4m+2)(4m+3)(4m+4)(4m+5)}
% \binom{5m}{m}\Big)
% \\ &
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}
% -4m 
% \frac{4(24m+1)}{5(4m+1)}\binom{5m}{m}
% %\frac{24m+2}{2m}\Big(2 \binom{5m}{m-1}\Big) -\frac{3}{2}\binom{5m-1}{m}
\\
%%%%%%%%%%%%%%% NEW %%%%%%%%%%%%%%%%%
%=&-\frac{8m(1536 m^6-355520 m^5+52672 m^4+3916 m^3-290 m^2+10 m+1)}
%{(5m-1)(4m+1)(2m+1)(4m+3)(m+1)(4m+5)}\binom{5m}{m}\\
%=&-\frac{8m(24m+1)(64m^5-14816m^4+2812m^3+46m^2-14m+1)}
%{(5m-1)(4m+1)(2m+1)(4m+3)(m+1)(4m+5)}\binom{5m}{m}.
=&-\frac{64(24m+1)f(m)}
{(5m-1)(4m+2)(4m+3)(4m+4)(4m+5)}\binom{5m}{m-1},
\end{align*}
%% B gave, H verified
where 
\[
f(m)=64m^5-14816m^4+2812m^3+46m^2-14m+1.
\]
%has four solutions consisting of real numbers.
%The four solutions are smaller than $232$.
%Hence, $b_{m+1}$ is negative for $m \ge 232$.
%% B gave, H verified


% Then we verified 
% that the coefficient of $y^{4m+8}$ in 
% the weight enumerator of 
% a singly even self-dual $[24m+2,12m+1,4m+2]$
% code with minimal shadow
% is negative for $155 \le m \le 232$.

\begin{theorem}
All coefficients in the weight enumerators of 
a singly even self-dual $[24m+2,12m+1,4m+2]$ code with minimal shadow
and its shadow are nonnegative integers 
if and only if $1 \le m \le 154$.
In particular,
for $m \ge 155$, there is no 
singly even self-dual $[24m+2,12m+1,4m+2]$
code with minimal shadow.
\end{theorem}
\begin{proof}
We verified that the equation 
%1536 m^6-355520 m^5+52672 m^4+3916 m^3-290 m^2+10 m+1=0
%64 m^5-14816 m^4+2812 m^3+46 m^2-14 m+1=0
$f(m)=0$
has three solutions consisting of real numbers and
the largest solution is in the interval $(231,232)$.
Thus, $b_{m+1}$ is negative for $m \ge 232$.
% there is no 
% singly even self-dual $[24m+2,12m+1,4m+2]$
% code with minimal shadow for $m \ge 232$.
% 
Using~\eqref{eq:WC} and~\eqref{eq:WS},
we determined numerically 
the weight enumerators of 
a singly even self-dual $[24m+2,12m+1,4m+2]$
code with minimal shadow and its shadow
for $m \le 231$.
The theorem follows from this calculation.
\end{proof}


%% As an example, we list the coefficient $A_{4m+8}$ of $y^{4m+8}$
%% in Table~\ref{Tab:co:24m+2} for $154 \le m \le 157$.
%% 
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \begin{table}[thb]
%% \caption{Coefficients $A_{4m+8}$}
%% \label{Tab:co:24m+2}
%% \begin{center}
%% %{\small
%% {\footnotesize
%% %{\scriptsize
%% \begin{tabular}{c|l}
%% \noalign{\hrule height0.8pt}
%% $m$ &  \multicolumn{1}{c}{$A_{4m+8}$} \\
%% \hline
%% 154&
%% $4890186454197915802986247894819457951256674504407720898992595618971$\\&
%% $2270934594781397281393190445592227712995673035218400230809320487148$\\&
%% $76867337674966924744997565338280000$\\
%% \hline
%% 155&
%% $-206176142471504065737150196897726175377603311491376866499241833664$\\&
%% $6692250419553935651367230154742905492780777724526933023005431546708$\\&
%% $06193434854146453536349680171424513920$\\
%% 156&
%% $-574411187620573057519087497149729810721148838921895088491983386086$\\&
%% $3270584651673056706055984535323149363921057930003595257373652115898$\\&
%% $433682868021743890843562399341957143680$\\
%% 157&
%% $-109295288138946379303030267312804412724149617085036971847732362698$\\&
%% $7940130001471022614754601962652470567193697440292835586348859473815$\\&
%% $03474875867028727668566993309791273903360$\\
%% %158&
%% %$-180975999779704020156191303031026056727186875468737769793229310405$\\&
%% %$6928443607350877950255076034531662906349310260232213650059703243564$\\&
%% %$232867217491753271690357059279647272465832$\\
%% %159&
%% %$-278654623910548300898590718413172850479740010603781875088087768364$\\&
%% %$8580654853235092695453154543458743074224531348560794713739394923293$\\&
%% %$8578287059511403007365248911215200134278400$\\
%% %160&
%% %$-410249647190330120571404592837344146944617262391899650536467937904$\\&
%% %$1552345816555834876159150251998128237977745725267932663058900099768$\\&
%% %$07496708312865118592506362799491987161603200$\\
%% \noalign{\hrule height0.8pt}
%% \end{tabular}
%% }
%% \end{center}
%% \end{table}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%
\section{Singly even
self-dual $[24m+4,12m+2,4m+2]$ codes with minimal shadow}\label{sec:24m+4}

\begin{proposition}
The weight enumerator of a singly even self-dual $[24m+4,12m+2,4m+2]$
code with minimal shadow is uniquely determined for each length.
\end{proposition}
\begin{proof}
The weight enumerator of a singly even self-dual 
$[4,2,2]$ code is uniquely determined.
Suppose that $m \ge 1$.
Let $C$ be a singly even self-dual $[24m+4,12m+2,4m+2]$ code
with minimal shadow.
The weight enumerators of $C$ and its shadow $S$ are written 
as using~\eqref{eq:WC} and~\eqref{eq:WS}.
% From \eqref{eq:ci}, \eqref{eq:ai} and \eqref{eq:bi},
% we have the following:
% \begin{equation}\label{eq:ci:24m+4}
% c_i=
% \begin{cases} 
% \alpha_{i,0} & \text{ if } i=0,1,\ldots,2m, \\
% \beta_{i,0}  & \text{ if } i=2m+1,2m+2,\ldots,3m. 
% \end{cases}
% \end{equation}
% Hence, $c_i$ $(i=0,1,\ldots,3m)$ 
Since $\alpha_{i,0}$ $(i=0,1,\ldots,2m)$ and
$\beta_{i,0}$ $(i=2m+1,2m+2,\ldots,3m)$ are calculated by
\eqref{eq:alpha} and~\eqref{eq:beta}, respectively,
from~\eqref{eq:c_i-all}, 
$c_i$ $(i=0,1,\ldots,3m)$ depends only on $m$.
This means that the weight enumerator of $C$ is uniquely
determined for each length.
\end{proof}

From~\eqref{eq:alpha},
\begin{align*}
\alpha_{2m,0}=&
% -\frac{24m+4}{4m}[\text{coeff.\ of } y^{2m-1} \text{ in }
% % (1+y)^{-12m-2-1+8m}(1-y)^{-4m}].
% (1+y)^{-4m-3}(1-y)^{-4m}]\\
% =&
% \frac{24m+4}{4m}\Big(3\binom{5m+1}{m-1}+\binom{5m}{m-2}\Big)
\frac{6m+1}{m}\left(3\binom{5m+1}{m-1}+\binom{5m}{m-2}\right)
\\=&
\frac{(6m+1)(8m+1)}{m(2m+1)}\binom{5m}{m-1}.
\end{align*}
From~\eqref{4d1},
\[
\beta_{2m,m}=\frac{1}{2^2}.   % 2^{-2}.
\]
From~\eqref{eq:beta},
\begin{align*}
\beta_{2m,0}
%=&2^{-12m-2+12m}\frac{3m}{2m}\binom{3m+2m-1}{3m-2m}
=\frac{1}{2^2}\frac{3}{2}\binom{5m-1}{m}
=\frac{3(4m+1)}{10m}\binom{5m}{m-1}.
\end{align*}
% %According to~\cite[p.~135]{Rains}, 
% Since
% \begin{align*}
% (1+y)^{-4m-3}(1-y)^{-4m} = (1-y^2)^{-4m-3}(1-y)^{3},
% \end{align*}
From~\eqref{eq:1}, 
\begin{align*}
b_m=&\frac{\alpha_{2m,0}-\beta_{2m,0}}{\beta_{2m,m}}  
%=&  
%\frac{24m+4}{m}\Big(3\binom{5m+1}{m-1}+\binom{5m}{m-2}\Big)
%-\frac{3}{2}\binom{5m-1}{m} \notag \\
=
%\binom{5m-1}{m}\frac{(12m+1)(38m+7)}{2(2m+1)(4m+1)}.
\frac{2(12m+1)(38m+7)}{5m(2m+1)}\binom{5m}{m-1}.
\end{align*}
\begin{remark}
Unfortunately, $b_m$ was incorrectly  reported
in~\cite{ZMFG}. The correct formula for $b_m$ is given in~\cite{ZMFG2}. 
We showed that $b_m$ is always a positive integer (see~\cite{ZMFG2}).
\end{remark}

% bm
% 
% \begin{proposition}
% The rational number $b_m$ in \eqref{eq:bm2} is a positive integer
% for all $m\geq1$.
% \end{proposition}
% \begin{proof}
% By a direct calculation, $b_1=78$ and $b_2=830$.
% Suppose that $m \ge 3$.
% We have 
% \begin{align*}
% &\binom{5m-1}{m}\frac{(12m+1)(38m+7)}{2(2m+1)(4m+1)}
% \\&=
% \binom{5m-1}{m}\frac12\left(57-\frac{10(22m+5)}{(2m+1)(4m+1)}\right)
% \nexteq
% \binom{5m-1}{m}\frac{57}{2}-\binom{5m-1}{m}\frac{5(22m+5)}{(2m+1)(4m+1)}
% \nexteq
% \binom{5m-1}{m}\frac{57}{2}-
% \frac{10(5m-1)\cdots(4m)(22m+5)}{(m!)(4m+2)(4m+1)}
% \nexteq
% \binom{5m-1}{m}\frac{57}{2}-
% \frac{40(5m-1)\cdots(4m+3)(22m+5)}{(m-3)!(m-1)(m-2)}
% \nexteq
% \binom{5m-1}{m}\frac{57}{2}-\binom{5m-1}{m-3}
% \frac{40(22m+5)}{(m-1)(m-2)}
% \nexteq
% \binom{5m-1}{m}\frac{57}{2}-\binom{5m-1}{m-3}
% 40\left(\frac{22}{m-2}+\frac{27}{(m-1)(m-2)}\right)
% \nexteq
% \binom{5m-1}{m}\frac{57}{2}
% -\binom{5m-1}{m-3}\frac{880}{m-2}
% -\binom{5m-1}{m-3}\frac{1080}{(m-1)(m-2)}.
% \end{align*}
% Note that $\binom{5m-1}{m}$ is even by Lucas's theorem.
% Note also that
% \[\binom{5m-1}{m-3}\frac{10}{m-2}\]
% is the $4$-ballot number $T_4(m-2,4m+1)$, where the $m$-ballot
% number is defined by (see~\cite[Sect.~2.6]{MR2777360})
% \[T_m(a,b)=\frac{b-ma+1}{b+a+1}\binom{b+a+1}{a}.\]
% Indeed,
% \begin{align*}
% T_4(m-2,4m+1)&=\frac{(4m+1)-4(m-2)+1}{(4m+1)+(m-2)+1}\binom{(4m+1)+(m-2)+1}
% {m-2}
% \nexteq
% \frac{10}{5m}\binom{5m}{m-2}
% \nexteq
% \frac{10}{m-2}\binom{5m-1}{m-3}.
% \end{align*}
% So it remains to show
% \[
% \binom{5m-1}{m-3}\frac{1080}{(m-1)(m-2)}
% \]
% is an integer.
% Now, we have 
% \[
% \binom{5m-1}{m-3}\frac{30}{(m-1)(m-2)}
% =\frac{6}{(4m+1)(4m+2)}\binom{5m}{m}.
% \]
% In addition, it is known that
% $\frac{6}{(4m+1)(4m+2)}\binom{5m}{m}$ is an integer~\cite{Schaeffer}
% (see also~\cite{OEIS}).
% This completes the proof.
% \end{proof}
% 
% \begin{remark}
% It was claimed in~\cite[Theorem 18, part 2]{ZMFG} that
% there is no singly even self-dual $[24m+4,12m+2,4m+2]$
% code with minimal shadow if 
% $b_m$ is not an integer (see also~\cite{ZMFG2}).
% However, the above proposition shows that
% $b_m$ is always a positive integer.
% This means that~\cite[Theorem 18, part 2]{ZMFG} is vacuous.
% \end{remark}


% Hence, we have
% \begin{equation}\label{eq:22}
% \alpha_{2m-1,0}= \beta_{2m-1,0} + \beta_{2m-1,m}b_m +\beta_{2m-1,m+1}b_{m+1}.
% \end{equation}
From~\eqref{eq:alpha}, 
\begin{align*}
\alpha_{2m-1,0}=&
% -\frac{24m+4}{4m-2}[\text{coeff.\ of } y^{2m-2} \text{ in }
% % (1+y)^{-12m-2-1+8m-4}(1-y)^{-4m+2}].
% (1+y)^{-4m-7}(1-y)^{-4m+2}]\\
% =&
% -\frac{24m+4}{4m-2}
-\frac{12m+2}{2m-1}
\left(
\binom{5m+5}{m-1}
+36\binom{5m+4}{m-2}
+126\binom{5m+3}{m-3}\right.
\\ &
\left.
+84\binom{5m+2}{m-4}
+9\binom{5m+1}{m-5}
%\Big).
\right)
\\=&
-\frac{16 (5m+1)(6m+1)(8m+1)(68m^2-m+3)}
{(4m+2)(4m+3)(4m+4)(4m+5)(4m+6)}\binom{5m}{m-1}.
\end{align*}
From~\eqref{4d1},
\[
\beta_{2m-1,m+1}=-\frac{1}{2^8}.  %-2^{-8}.
\]
From~\eqref{eq:beta}, 
\begin{align*}
\beta_{2m-1,0}
%=&(-1)2^{-12m-2+12m-6}\frac{3m}{2m-1}\binom{3m+2m-1-1}{3m-2m+1}\\
=&-\frac{1}{2^{8}}\frac{3m}{2m-1}\binom{5m-2}{m+1} 
=-\frac{1}{2^{5}}
\frac{3(4m-1)(4m+1)}{5(5m-1)(m+1)}\binom{5m}{m-1}, \\
\beta_{2m-1,m}
%=&(-1)2^{-12m-2+12m-6}\frac{3m-m}{2m-1}\binom{3m+2m-1-m-1}{3m-2m+1-m} \\
%=&-\frac{1}{2^8}\frac{2m}{2m-1}\binom{4m-2}{1}\\
=&-\frac{m}{2^6}.
\end{align*}
% %According to~\cite[p.~135]{Rains}, 
% Since
% \begin{align*}
% (1+y)^{-4m-7}(1-y)^{-4m+2} = (1-y^2)^{-4m-7}(1-y)^{9}
% \end{align*}
% and 
% \[
% (1-y)^9=
% 1 - 9y + 36y^2 - 84y^3 + 126y^4 - 126y^5 + 84y^6 - 36y^7 + 9y^8 - y^9,
% \]
From~\eqref{eq:2}, 
\begin{align*}
b_{m+1} =&\frac{\alpha_{2m-1,0}- \beta_{2m-1,0} - 
\beta_{2m-1,m}b_m}{\beta_{2m-1,m+1}}
% \\ =&
% \frac{\alpha_{2m-1,0}
% +2^{-8}\frac{3m}{2m-1}\binom{5m-2}{m+1}
% +2^{-8} 4m b_m}{-\frac{1}{2^8}}       %{-2^{-8}}
% \\ =&
% (-2^8)\alpha_{2m-1,0}
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}- 4mb_m
% 
% \\ =&
% 2^8 \frac{24m+4}{4m-2}
% \Big(
% \binom{5m+5}{m-1}
% +36\binom{5m+4}{m-2}
% +126\binom{5m+3}{m-3}
% \\ &
% +84\binom{5m+2}{m-4}
% +9\binom{5m+1}{m-5}
% \Big)
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}-4mb_m
% \\ =&
% 2^8 \frac{24m+4}{4m-2}
% \Big(
% \binom{5m+5}{m-1}
% +36\binom{5m+4}{m-2}
% +126\binom{5m+3}{m-3}
% \\ &
% +84\binom{5m+2}{m-4}
% +9\binom{5m+1}{m-5}
% \Big)
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}
% \\ &
% -4m \Big(
% \binom{5m-1}{m}\frac{(12m+1)(38m+7)}{2(2m+1)(4m+1)}
% \Big)
% \\ =&
% 2^8 
% \Big(
% \frac{16 m (5m+1)(6m+1)(8m+1)(68m^2-m+3)}
% {(4m+1)(4m+2)(4m+3)(4m+4)(4m+5)(4m+6)}\binom{5m}{m}
% \Big)
% \\ &
% -\frac{3m}{2m-1}\binom{5m-2}{m+1}
% -4m 
% \binom{5m-1}{m}\frac{(12m+1)(38m+7)}{2(2m+1)(4m+1)}
% %%%%%%%%%%%%%%%%%%%%%
\\ 
=&
%-\frac{8m(12m+1)(1216 m^6-212096 m^5-33020 m^4+5440 m^3+1171 m^2+88 m+6)}
%{(5m-1)(4m+1)(2m+1)(4m+3)(m+1)(4m+5)(2m+3)}
%\binom{5m}{m}
-\frac{128(12m+1)f(m)}
{(5m-1)(4m+2)(4m+3)(4m+4)(4m+5)(4m+6)}
\binom{5m}{m-1},
\end{align*}
%% B gave, H verified
where
\[
f(m) =
1216 m^6-212096 m^5-33020 m^4+5440 m^3+1171 m^2+88 m+6.
\]



% Hence, $b_{m+1}$ is negative for $m \ge 175$.
%% B gave, H verified

\begin{theorem}
All coefficients in 
the weight enumerators of 
a singly even self-dual $[24m+4,12m+2,4m+2]$ code with minimal shadow
and its shadow are nonnegative integers 
if and only if $1 \le m \le 155$.
In particular, for $m \ge 156$, there is no 
singly even self-dual $[24m+4,12m+2,4m+2]$
code with minimal shadow.
\end{theorem}
\begin{proof}
We verified that the equation 
%1216 m^6-212096 m^5-33020 m^4+5440 m^3+1171 m^2+88 m+6=0
$f(m)=0$
has two solutions consisting of real numbers and
the largest solution is in the interval $(174,175)$.
Thus, $b_{m+1}$ is negative for $m \ge 175$.
% there is no 
% singly even self-dual $[24m+4,12m+2,4m+2]$
% code with minimal shadow for $m \ge 175$.
% 
Using~\eqref{eq:WC} and~\eqref{eq:WS},
we determined numerically 
the weight enumerators of 
a singly even self-dual $[24m+4,12m+2,4m+2]$
code with minimal shadow and its shadow
for $m \le 174$.
The theorem follows from this calculation.
\end{proof}


%% As an example, we list the coefficient $A_{4m+8}$ of $y^{4m+8}$
%% in Table~\ref{Tab:co:24m+4} for $155 \le m \le 158$.
%% 
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \begin{table}[thb]
%% \caption{Coefficients $A_{4m+8}$}
%% \label{Tab:co:24m+4}
%% \begin{center}
%% %{\small
%% {\footnotesize
%% %{\scriptsize
%% \begin{tabular}{c|l}
%% \noalign{\hrule height0.8pt}
%% $m$ &  \multicolumn{1}{c}{$A_{4m+8}$} \\
%% \hline
%% 155&
%% $7332600665882768402877081434904332007301587785745398205466783265470$\\&
%% $1671004540956124436039154155743838640267599303956596585995401468043$\\&
%% $283443551436293252551654062870376896$\\
%% \hline
%% 156&
%% $-111769743737893023452325039879158493898426425169431015924580546673$\\&
%% $2163653085959261838615625088248664552782159460891657493693398486860$\\&
%% $183262490740151280080571364863277515648$\\
%% 157&
%% $-380759393689034008724278992606202795758722337585139347195449534594$\\&
%% $5653340520327599485552813273870705303474214871952564779073673741487$\\&
%% $1127266014107572642553137287945214833040$\\
%% 158&
%% $-761374552460994442116438538371986918555032983206212084842001552633$\\&
%% $3668744410012575241624642663190692935434768986756421998096789663725$\\&
%% $71224850561794257415496645868975282511016$\\
%% %159&
%% %$-128944281998451693916326992585693717462169051289988138373688320723$\\&
%% %$1342523861940306390486125205870967835249273550985481340506075874424$\\&
%% %$8054631699264625354694295429964265152289408$\\
%% %160&
%% %$-201109853258702697754395332374568753940097321421885741229693668100$\\&
%% %$8959736564678334795384303231796265710422615068069516000447882690646$\\&
%% %$22910007150542792694315955849459296391367040$\\
%% \noalign{\hrule height0.8pt}
%% \end{tabular}
%% }
%% \end{center}
%% \end{table}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%




%%%%%%%%%%%%%%%%
\section{Singly even
self-dual $[24m+10,12m+5,4m+2]$ codes with minimal shadow}\label{sec:24m+10}


\begin{lemma}[Harada~\cite{H60}]
\label{lem:H}
Suppose that $n \equiv 2 \pmod 8$.
Let $C$ be a singly even self-dual $[n,n/2,d]$ code
with minimal shadow.
If $d \equiv 2 \pmod 4$, then $a_{d/2} =b_{(d-2)/4}$.
\end{lemma}

As a consequence, the weight enumerator of a singly even
self-dual $[58,29,10]$ code with minimal shadow
was uniquely determined in~\cite{H60}.  

\begin{proposition}
The weight enumerator of a singly even self-dual 
$[24m+10,12m+5,4m+2]$ code with minimal shadow
is uniquely determined for each length.
\end{proposition}
\begin{proof}
The weight enumerator of a singly even self-dual 
$[10,5,2]$ code with minimal shadow is uniquely determined.
Suppose that $m \ge 1$.
Let $C$ be a singly even self-dual $[24m+10,12m+5,4m+2]$ code 
with minimal shadow.
The weight enumerators of $C$ and its shadow $S$ are written 
as in~\eqref{eq:WC} and~\eqref{eq:WS}, respectively.
Since $\alpha_{i,0}$ $(i=0,1,\ldots,2m)$ and
$\beta_{i,0}$ $(i=2m+2,2m+3,\ldots,3m+1)$ are calculated 
by~\eqref{eq:alpha} and~\eqref{eq:beta}, respectively,
from~\eqref{eq:c_i-all}, 
$c_i$ $(i=0,1,\ldots,2m,2m+2,\ldots,3m+1)$ depends only on $m$.
% From \eqref{eq:ci}, \eqref{eq:ai} and \eqref{eq:bi},
% we have the following:
% \begin{equation}\label{eq:ci:24m+10}
% c_i=
% \begin{cases} 
% \alpha_{i,0} & \text{ if } i=0,1,\ldots,2m, \\
% \beta_{i,0}  & \text{ if } i=2m+2,2m+3,\ldots,3m+1. 
% \end{cases}
% \end{equation}
% c_{2m+1} =& \sum_{j=0}^{2m+1} \alpha_{2m+1,j}a_j
% =\alpha_{2m+1,0}+\alpha_{2m+1,2m+1}a_{2m+1},
% \\
% c_{2m+1}
% =&\sum_{j=0}^m \beta_{2m+1,j}b_j =\beta_{2m+1,0}+\beta_{2m+1,m}b_m.

From~\eqref{4b1} and~\eqref{4d1}, we have
\[
\alpha_{2m+1,2m+1}=1 \text{ and }
\beta_{2m+1,m}=-2,
\]
respectively.
By Lemma~\ref{lem:H}, it holds that $a_{2m+1}=b_m$.
From~\eqref{eq:4}, we obtain
\begin{equation}\label{eq:a2m+1}
%a_{2m+1}(\alpha_{2m+1,2m+1}-\beta_{2m+1,m}) =
%\beta_{2m+1,0}-\alpha_{2m+1,0}.
a_{2m+1}=\frac{\beta_{2m+1,0}-\alpha_{2m+1,0}}{3}.
\end{equation}
Therefore, from~\eqref{eq:4}, 
$c_{2m+1}$ depends only on $m$.
This means that the weight enumerator of $C$ is uniquely
determined for each length.
\end{proof}

From~\eqref{eq:alpha},
we have
\begin{align*}
\alpha_{2m+1,0}=&
% -\frac{24m+10}{4m+2}[\text{coeff.\ of } y^{2m} \text{ in }
% % (1+y)^{-12m-5-1+8m+4}(1-y)^{-4m-2}].
% (1+y)^{-4m-2}(1-y)^{-4m-2}]\\
% =&-\frac{24m+10}{4m+2}\binom{4m+1+m}{m}\\
-\frac{12m+5}{2m+1}\binom{5m+1}{m}.
\end{align*}
From~\eqref{eq:beta}, we have
\begin{align*}
\beta_{2m+1,0}
%&=(-1)^{2m+1} 2^{-(12m+5)+12m+6}
%\frac{3m+1}{2m+1}\binom{(3m+1)+2m+1-1}{(3m+1)-2m-1}\\ &
= -2 \frac{3m+1}{2m+1}\binom{5m+1}{m}.
\end{align*}
% Hence, 
% \begin{align}\label{eq:ne0}
% \beta_{2m+1,0}-\alpha_{2m+1,0}
% %=&-\frac{6m+2}{2m+1}\binom{5m+1}{m}+\frac{12m+5}{2m+1}\binom{5m+1}{m}\\
% %=& \frac{12m+5-6m-2}{2m+1}\binom{5m+1}{m}\\
% =\frac{6m+3}{2m+1}\binom{5m+1}{m}
% =3\binom{5m+1}{m} \ne 0.
% \end{align}
% This implies that $\alpha_{2m+1,2m+1} \ne \beta_{2m+1,m}$.
% Then
% \begin{equation}\label{eq:a2m+1}
% a_{2m+1}=\frac{\beta_{2m+1,0}-\alpha_{2m+1,0}}
%{\alpha_{2m+1,2m+1}-\beta_{2m+1,m}}.
% \end{equation}
Since $a_{2m+1}=b_m$, 
from~\eqref{eq:a2m+1}, we have 
\[
b_m=\binom{5m+1}{m}=\frac{5m+1}{4m+1}\binom{5m}{m}.  
\]
%
% \begin{lemma}
% $\alpha_{2m+1,2m+1}=1$.
% \end{lemma}
% \begin{proof}{\bf (Stefka)}
% (We are looking for $\kappa_{2m+1,2m+1}$, which in our case is denoted by 
% $\alpha_{2m+1,2m+1}$. 
% Perhaps I have to add some more reasons about that? 
% Moreover, it seems to me that I have seen a calculation for this coefficient 
% but I didn't find such in the papers I looked at).
% 
% By the B\"{u}rmann--Lagrange theorem (see~\cite{Rains}), we have
% \[
% \alpha_{2m+1,2m+1}=
% \frac{1}{2m+1}[\text{coeff.\ of } y^{2m} \text{ in } 
% ((2m+1)y^{2m}f(y) +y^{2m+1}f'(y))(\frac{y}{g(y)})^{2m+1}],
% \]
% where
% $g(y)=\frac{y(1-y)^2}{(1+y)^4} \text{ and }
% 64m^5-14816m^4+2812m^3+46m^2-14m+164m^5-14816m^4+2812m^3+46m^2-14m+1
% f(y)=(1+y)^{-n/2}=(1+y)^{-12m-5}$% .
% Since
% \[
% (\frac{y}{g(y)})^{2m+1}=(\frac{(1+y)^4}{(1-y)^2})^{2m+1}
% =(1-y)^{-4m-2}(1+y)^{8m+4},
% \]
% we have
% $\alpha_{2m+1,2m+1}=\frac{2m+1}{2m+1}=1$.
% \end{proof}
%
%
% bm
%
% \begin{proposition}
% $b_m=a_{2m+1}=\binom{5m+1}{m}$.
% \end{proposition}
% \begin{proof}
% From \eqref{eq:beta},
% \[
% \beta_{2m+1,m}=-2^{-12m-5+12m+6}\frac{3m+1-m}{2m+1}
% \binom{3m+1+2m+1-m-1}{3m+1-2m-1-m}
% =-2.
% \]
% The result follows from \eqref{eq:ne0}, \eqref{eq:a2m+1} and the above lemma.
% \end{proof}
%
% Hence, we have
% \[
% b_{m+1}=\frac{\alpha_{2m,0}-\beta_{2m,0}-\beta_{2m,m}a_{2m+1}}{\beta_{2m,m+1}}.
% \]
From~\eqref{eq:alpha},
\begin{align*}
\alpha_{2m,0}=&
% -\frac{24m+10}{4m}[\text{coeff.\ of } y^{2m-1} \text{ in }
% (1+y)^{-4m-6}(1-y)^{-4m}]\\
% =&
% \frac{24m+10}{4m}\Big(
\frac{12m+5}{2m}\left(
6\binom{5m+4}{m-1}
+20\binom{5m+3}{m-2}
+6\binom{5m+2}{m-3}
\right)
\\
=&
\frac{4(12m+5)(5m+1)(5m+2)(32m^2 +19m+3)}{
(4m+1)(4m+2)(4m+3)(4m+4)(4m+5)}\binom{5m}{m}.
\end{align*}
From~\eqref{4d1},
\[
\beta_{2m,m+1}=\frac{1}{2^5}.   %   2^{-5}.
\]
From~\eqref{eq:beta},
\begin{align*}
\beta_{2m,0}
%=&2^{-(12m+5)+12m} \frac{3m+1}{2m} \binom{3m+1+2m-1}{3m+1-2m} \\
=&\frac{1}{2^5}\frac{3m+1}{2m} \binom{5m}{m+1}
=\frac{1}{2^4}\frac{3m+1}{m+1} \binom{5m}{m},
\\
\beta_{2m,m}
%=&2^{-(12m+5)+12m}\frac{3m+1-m}{2m} \binom{3m+1+2m-m-1}{3m+1-2m-m} \\
=&\frac{1}{2^5} \frac{2m+1}{2m} 4m
=\frac{2m+1}{2^4}.
\end{align*}
% Since
% \begin{align*}
% (1+y)^{-4m-6}(1-y)^{-4m} = (1-y^2)^{-4m-6}(1-y)^{6}
% \end{align*}
% and
% \begin{align*}
% (1-y)^6=y^6- 6y^5 + 15y^4 - 20y^3 + 15y^2 - 6y +1,
% \end{align*}
From~\eqref{eq:3},
\begin{align*}
b_{m+1}=&\frac{\alpha_{2m,0}-\beta_{2m,0}-\beta_{2m,m}b_{m}}{\beta_{2m,m+1}}\\
% =&
% 2^5\Big(
% \frac{12m+5}{m}\Big(
% 3\binom{5m+4}{m-1}
% +10\binom{5m+3}{m-2}
% +3\binom{5m+2}{m-3}
% \Big)
% \\&
% -\frac{1}{2^5} \frac{3m+1}{2m} \binom{5m}{m+1}
% -\frac{1}{2^4} (2m+1)b_{m}
% \Big)
% \\
% =&
% 2^5
% \frac{12m+5}{m}\Big(
% 3\binom{5m+4}{m-1}
% +10\binom{5m+3}{m-2}
% +3\binom{5m+2}{m-3}
% \Big)
% \\&
% -\frac{3m+1}{2m} \binom{5m}{m+1}
% -2(2m+1)b_{m}
% \\
% %=&
% %\frac{- 5120 m^6+ 1201152 m^5 + 1924160 m^4 + 1211072 m^3 + 374448 m^2 + 56752 m +3360 }
% %{(4m+5)(4m+4)(4m+3)(4m+2)(4m+1)}
% %\binom{5m}{m}
% %\\
% =&
% 2^5
% \frac{4(12m+5)(5m+1)(5m+2)(32m^2 +19m+3)}{
% (4m+1)(4m+2)(4m+3)(4m+4)(4m+5)}\binom{5m}{m}
% \\&
% -\frac{3m+1}{2m} \binom{5m}{m+1}
% -2(2m+1) \binom{5m+1}{m}
% \\
=&
-\frac{16(5m+2)f(m)}
{(4m+1)(4m+2)(4m+3)(4m+4)(4m+5)}
\binom{5m}{m},
\end{align*}
where
\[
 f(m)=
64m^5- 15040m^4- 18036m^3 - 7924m^2 - 1511m -105.
\]

\begin{theorem}
All coefficients in 
the weight enumerators of 
a singly even self-dual $[24m+10,12m+5,4m+2]$ code with minimal shadow
and its shadow are nonnegative integers 
if and only if $1 \le m \le 159$.
In particular, for $m \ge 160$, there is no 
singly even self-dual $[24m+10,12m+5,4m+2]$
code with minimal shadow.
\end{theorem}
\begin{proof}
We verified that the equation 
%64m^5- 15040m^4- 18036m^3 - 7924m^2 - 1511m -105=0
$f(m)=0$
has three solutions consisting of real numbers and
the largest solution is in the interval $(236,237)$.
%Hence, $b_{m+1}$ is negative for $m \ge 237$.
Thus, $b_{m+1}$ is negative for $m \ge 237$.
% there is no 
% singly even self-dual $[24m+10,12m+5,4m+2]$
% code with minimal shadow for $m \ge 237$.
% 
Using~\eqref{eq:WC} and~\eqref{eq:WS},
we determined numerically 
the weight enumerators of 
a singly even self-dual $[24m+10,12m+5,4m+2]$
code with minimal shadow and its shadow
for $m \le 236$.
The theorem follows from this calculation. 
\end{proof}


%% {\color{red}
%% As an example, we list the coefficient $A_{4m+8}$ of $y^{4m+8}$
%% in Table~\ref{Tab:co:24m+10} for $159 \le m \le 162$.
%% }
%% 
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% \begin{table}[thb]
%% \caption{Coefficients $A_{4m+8}$}
%% \label{Tab:co:24m+10}
%% \begin{center}
%% %{\small
%% {\footnotesize
%% %{\scriptsize
%% \begin{tabular}{c|l}
%% \noalign{\hrule height0.8pt}
%% $m$ &  \multicolumn{1}{c}{$A_{4m+8}$} \\
%% \hline
%% 159&
%% $5525331578534582614839797242844269598144254764673468563174043700948$\\&
%% $2428795908507558859755200436107314254406733163870584120973577215701$\\&
%% $1343444278017813481041789675440572586240$\\
%% \hline
%% 160&
%% $-121018235937005407437400946680774181638246251362508521394468564716$\\&
%% $7653081188900770512737704959867010246265326572161367433285916138804$\\&
%% $9670667279515793001061572305246153312228390$\\
%% 161&
%% $-376483486321071738236192647067479649643410795434276231081904389646$\\&
%% $8537426575086823675696022601964992687103167970143272028013605159839$\\&
%% $42086673291136535483388336674273405519108612$\\
%% 162&
%% $-737244937151683548044556436013537079045023922752428530156464919565$\\&
%% $8650660301324676164398467181636405574401806400417875586666842281806$\\&
%% $259463178773519642385384365487380650603228822$\\
%% \noalign{\hrule height0.8pt}
%% \end{tabular}
%% }
%% \end{center}
%% \end{table}
%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



%%%%%%%%%%%%%%%%
\section{Remaining cases}\label{sec:rem}

For the remaining cases, we demonstrate that
the weight enumerator of a singly even self-dual 
code with minimal shadow is not uniquely determined.


\subsection{Singly even
self-dual $[24m+6,12m+3,4m+2]$ codes with minimal shadow}



Using~\eqref{eq:WC} and~\eqref{eq:WS},
the possible weight enumerators of 
a singly even self-dual $[30,15,6]$ code with minimal shadow
and its shadow are given by
\begin{align*}
&
1+( 35  - 8 \beta) y^6 
+( 345 + 24 \beta) y^8 
+ 1848 y^{10} 
+ \cdots,
\\&
\beta y^3 
+( 240 - 6 \beta) y^7 
+( 6720 + 15 \beta) y^{11} 
+ \cdots,
\end{align*}
respectively, where $\beta$ is an integer with $1 \le \beta \le 4$.
It is known that there is 
a singly even self-dual $[30,15,6]$ code with minimal shadow
for $\beta \in \{1,2\}$ (see~\cite{C-S}).

Using~\eqref{eq:WC} and~\eqref{eq:WS},
the possible weight enumerators of 
a singly even self-dual $[54,27,10]$ code with minimal shadow
and its shadow are given by
\begin{align*}
&
1 
+( 351  - 8 \beta) y^{10} 
+( 5543  + 24 \beta) y^{12} 
+( 43884  + 32 \beta) y^{14} 
+ \cdots,
\\&
y^3 
+(- 12 + \beta) y^{7 }
+( 2874 - 10 \beta) y^{11} 
+( 258404 + 45 \beta) y^{15} 
+ \cdots,
\end{align*}
respectively, where $\beta$ is an integer with $12 \le \beta \le 43$.
It is known that there is 
a singly even self-dual $[54,27,10]$ code with minimal shadow
for $\beta \in \{12,13,\ldots,20,21,22,24,26\}$ (see~\cite{YL14}).




%%%%%%%%%%%%%%%%
\subsection{Singly even
self-dual $[24m+22,12m+11,4m+4]$ codes with minimal shadow}

Using~\eqref{eq:WC} and~\eqref{eq:WS},
the possible weight enumerators of 
a singly even self-dual $[22,11,4]$ code with minimal shadow
and its shadow are given by
\begin{align*}
&
1 
+ 2 \beta y^4 
+( 77  - 2 \beta) y^6 
+( 330  - 6 \beta) y^8 
+( 616  + 6 \beta) y^{10} 
+ \cdots,
\\&
\beta y^3 
+( 352  - 4 \beta) y^7 
+( 1344  + 6 \beta) y^{11} 
+ \cdots,
\end{align*}
respectively, where $\beta$ is an integer with $1 \le \beta \le 38$.
It is known that there is a 
singly even self-dual $[22,11,4]$ code with minimal shadow
for $\beta \in \{2,4,6,8,10,14\}$ (see~\cite{PS75}).
 
Using~\eqref{eq:WC} and~\eqref{eq:WS},
the possible weight enumerators of 
a singly even self-dual $[46,23,8]$ code with minimal shadow
and its shadow are given by
\begin{align*}
&
1 
+ 2 \beta y^8 
+ (884 - 2 \beta) y^{10} 
+ (10556 - 14 \beta) y^{12} 
+ (54621  + 14 \beta) y^{14} 
+ \cdots,
\\&
y^3 
+(- 10  + \beta) y^7 
+( 6669  - 8 \beta) y^{11} 
+( 242760  + 28 \beta) y^{15} 
+ \cdots,
\end{align*}
respectively, where $\beta$ is an integer with $10 \le \beta \le 442$.
Let $C_{46}$ be the code with generator matrix 
$\left[
\begin{array}{cc}
I_{23} & R
\end{array}
\right]$,
where $I_{23}$ denotes the identity matrix of order $23$ and
$R$ is the $23 \times 23$ circulant matrix with
first row
\[
(01011101011100000111110). 
\]
We verified that $C_{46}$ is a singly even self-dual
$[46,23,8]$ code.
By considering self-dual neighbors of $C_{46}$,
we found  singly even self-dual
$[46,23,8]$ codes $N_{46,i}$ with minimal shadow $(i=1,2,\ldots,10)$.
These codes are constructed as
$\langle C_{46} \cap \langle x \rangle^\perp, x \rangle$,
where the supports $\supp(x)$ of $x$ are listed in Table~\ref{Tab:nei}.
The values $\beta$ in the weight enumerators of $N_{46,i}$
are also listed in the table.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{table}[!h]
  \begin{center}
\begin{tabular}{c|l|c}
\noalign{\hrule height0.8pt}
Codes & \multicolumn{1}{c|}{$\supp(x)$} & $\beta$\\
\hline
$N_{46,1}$ & $\{1,24,26,27,29,30,31,32,33,34,36,37,42,43,45,46\}$&36\\
$N_{46,2}$ & $\{1,27,28,31,33,35,36,37,42,43,45,46\}$&42\\
$N_{46,3}$ & $\{10,11,20,27,29,34,38,41,42,45\}$&44\\
$N_{46,4}$ & $\{5,6,25,29,30,32,33,36,40,41,44,45\}$&46\\
$N_{46,5}$ & $\{1,23,28,29,30,31,32,37,40,41,44,45\}$&48\\
$N_{46,6}$ & $\{1,26,27,28,30,32,35,36,37,42,43,45\}$&50\\
$N_{46,7}$ & $\{2,3,24,25,26,28,29,33,34,36,37,41,42,44\}$&52\\
$N_{46,8}$ & $\{1,25,28,29,32,33,34,36,38,42,43,45\}$&54\\
$N_{46,9}$ & $\{1,23,24,27,30,36,40,41,44,45\}$&56\\
$N_{46,10}$& $\{1,2,25,29,30,33,35,38,44,46\}$&58\\
\noalign{\hrule height0.8pt}
\end{tabular}
  \end{center}
  \caption{\label{Tab:nei} Singly even self-dual $[46,23,8]$ codes $N_{46,i}$ with 
minimal shadows.}
\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%



The possible weight enumerators of 
a singly even self-dual $[70,35,12]$ code with minimal shadow
and its shadow are given by
\begin{align*}
&
1 
+ 2 \beta y^{12} 
+( 9682 - 2 \beta) y^{14} 
+( 173063 - 22 \beta) y^{16} 
%+( 1210792 + 22 \beta) y^{18} 
+ \cdots,
\\&
y^3 
+(- 104 + \beta )y^{11}
+( 88480 - 12 \beta) y^{15}
%+( 7363356 +  66 \beta) y^{19}
+ \cdots,
\end{align*}
respectively, where $\beta$ is an integer 
$104 \le \beta \le 4841$~\cite{H70}.
It is known that there is a 
singly even self-dual $[70,35,12]$ code with minimal shadow
for many different $\beta$~\cite[p.~1191]{YLGI}.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
The first author 
is supported by Grant DN 02/2/13.12.2016 of the 
Bulgarian National Science Fund.
The second author 
is supported by JSPS KAKENHI Grant Number 15H03633.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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