% EJC papers *must* begin with the following two lines. \documentclass[12pt]{article} \usepackage{e-jc} % Please remove all other commands that change parameters such as % margins or pagesizes. % only use standard LaTeX packages % only include packages that you actually need % we recommend these ams packages \usepackage{amssymb,amsthm,amsmath,latexsym} %\usepackage{amsthm,amsmath,amssymb} % we recommend the graphicx package for importing figures \usepackage{graphicx} \usepackage{url} % use this command to create hyperlinks (optional and recommended) \usepackage[colorlinks=true,citecolor=black,linkcolor=black,urlcolor=blue]{hyperref} % use these commands for typesetting doi and arXiv references in the bibliography \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{\texttt{doi:#1}}} \newcommand{\arxiv}[1]{\href{http://arxiv.org/abs/#1}{\texttt{arXiv:#1}}} % all overfull boxes must be fixed; % i.e. there must be no text protruding into the margins % declare theorem-like environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{fact}[theorem]{Fact} \newtheorem{observation}[theorem]{Observation} \newtheorem{claim}[theorem]{Claim} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{open}[theorem]{Open Problem} \newtheorem{problem}[theorem]{Problem} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \newtheorem{note}[theorem]{Note} \newcommand{\nexteq}{\displaybreak[0]\\ &=} \newcommand{\FF}{\mathbb{F}} \newcommand{\ZZ}{\mathbb{Z}} \newcommand{\RR}{\mathbb{R}} \newcommand{\allone}{\mathbf{1}} \DeclareMathOperator{\Aut}{Aut} \DeclareMathOperator{\wt}{wt} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\supp}{supp} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % if needed include a line break (\\) at an appropriate place in the title \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 $0\leq i3m+l-j$,}\\ (-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. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \bibliographystyle{plain} % \bibliography{myBibFile} % If you use BibTeX to create a bibliography % then copy and past the contents of your .bbl file into your .tex file \begin{thebibliography}{30} \bibitem{Magma}W. Bosma, J. Cannon, and C. Playoust. \newblock The Magma algebra system I: The user language. \newblock {\em J. Symbolic Comput.}, 24:235--265, 1997. \bibitem{performance} S. Bouyuklieva, A. Malevich, and W. Willems. \newblock On the performance of binary extremal self-dual codes. \newblock {\em Adv.\ Math.\ Commun.}, 5:267--274, 2011. \bibitem{BV}S. Bouyuklieva and Z. Varbanov. \newblock Some connections between self-dual codes, combinatorial designs and secret sharing schemes. \newblock {\em Adv.\ Math.\ Commun.}, 5:191--198, 2011. \bibitem{BW}S. Bouyuklieva and W. Willems. \newblock Singly even self-dual codes with minimal shadow. \newblock {\em IEEE Trans.\ Inform.\ Theory}, 58:3856--3860, 2012. \bibitem{C-S} J. H.~Conway and N. J. A.~Sloane. \newblock A new upper bound on the minimal distance of self-dual codes. \newblock {\em IEEE\ Trans.\ Inform.\ Theory}, 36:1319--1333, 1990. \bibitem{H70} M.~Harada. \newblock The existence of a self-dual $[70,35,12]$ code and formally self-dual codes. \newblock {\em Finite Fields Appl.}, 3:131--139, 1997. \bibitem{H60} M.~Harada. \newblock Binary extremal self-dual codes of length $60$ and related codes. \newblock {\em Des.\ Codes Cryptogr.}, (to appear), \arxiv{1706.01694}. %arXiv:~1706.01694. \bibitem{PS75} V. Pless and N. J. A. Sloane. \newblock On the classification and enumeration of self-dual codes. \newblock {\em J. Combin.\ Theory Ser.~A}, 18:313--335, 1975. \bibitem{Rains} E. M.~Rains. \newblock {Shadow bounds for self-dual codes}. \newblock {\em IEEE Trans.\ Inform.\ Theory}, 44:134--139, 1998. \bibitem{YL14}N. Yankov and M. H. Lee. \newblock New binary self-dual codes of lengths 50--60. \newblock {\em Des.\ Codes Cryptogr.}, 73:983--996, 2014. \bibitem{YLGI}N. Yankov, M. H. Lee, G\"{u}rel, and M. Ivanova. \newblock Self-dual codes with an automorphism of order $11$. \newblock {\em IEEE Trans.\ Inform.\ Theory}, 61:1188--1193, 2015. \bibitem{ZMFG}T. Zhang, J. Michel, T. Feng, and G. Ge. \newblock On the existence of certain optimal self-dual codes with lengths between $74$ and $116$. \newblock {\em Electron.\ J. Combin.}, 22:Paper 4.33 25~pp., 2015. \bibitem{ZMFG2}T. Zhang, J. Michel, T. Feng, and G. Ge. \newblock Erratum to ``On the existence of certain optimal self-dual codes with lengths between $74$ and $116$''. (submitted). \end{thebibliography} \end{document}