On the nonexistence of ternary extremal self-dual codes

Lengths of extremal square-free ternary words

A square-free word $w$ over a fixed alphabet $Σ$ is extremal if every word obtained from $w$ by inserting a single letter from $Σ$ (at any position) contains a square. Grytczuk et al. recently introduced the concept of extremal square-free word, and demonstrated that there are arbitrarily long extremal square-free ternary words. We find all lengths which admit an extremal square-free ternary word. In particular, we show that there is an extremal square-free ternary word of every sufficiently large length. We also solve the analogous problem for circular words.

三元极值四负循环自对偶码

Ternary extremal four-negacirculant self-dual codes

本文研究三元四负循环自对偶码，通过穷举搜索完成了长度40至60的极值码分类。

Some restrictions on the weight enumerators of near-extremal ternary self-dual codes and quaternary Hermitian self-dual codes

We give restrictions on the weight enumerators of ternary near-extremal self-dual codes of length divisible by $12$ and quaternary near-extremal Hermitian self-dual codes of length divisible by $6$. We consider the weight enumerators for which there is a ternary near-extremal self-dual code of length $12m$ for $m =3,4,5,6$. Also we consider the weight enumerators for which there is a quaternary near-extremal Hermitian self-dual code of length $6m$ for $m =4,5,6$.

Hadamard matrices of order $36$ formed by codewords in some ternary self-dual codes

In this note, we study the existence of Hadamard matrices of order $36$ formed by codewords of weight $36$ in some ternary near-extremal self-dual codes of length $36$.

A Complete Classification of Ternary Self-Dual Codes of Length 24

Ternary self-dual codes have been classified for lengths up to 20. At length 24, a classification of only extremal self-dual codes is known. In this paper, we give a complete classification of ternary self-dual codes of length 24 using the classification of 24-dimensional odd unimodular lattices.

The codes and the lattices of Hadamard matrices

It has been observed by Assmus and Key as a result of the complete classification of Hadamard matrices of order 24, that the extremality of the binary code of a Hadamard matrix H of order 24 is equivalent to the extremality of the ternary code of H^T. In this note, we present two proofs of this fact, neither of which depends on the classification. One is a consequence of a more general result on the minimum weight of the dual of the code of a Hadamard matrix. The other relates the lattices obtained from the binary code and from the ternary code. Both proofs are presented in greater generality …

Hadamard matrices related to a certain series of ternary self-dual codes

In 2013, Nebe and Villar gave a series of ternary self-dual codes of length $2(p+1)$ for a prime $p$ congruent to $5$ modulo $8$. As a consequence, the third ternary extremal self-dual code of length $60$ was found. We show that the ternary self-dual code contains codewords which form a Hadamard matrix of order $2(p+1)$ when $p$ is congruent to $5$ modulo $24$. In addition, it is shown that the ternary self-dual code is generated by the rows of the Hadamard matrix. We also demonstrate that the third ternary extremal self-dual code of length $60$ contains at least two inequivalent Hadamard matr…

Nonexistence of certain singly even self-dual codes with minimal shadow

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 $m \ge 155$, $m \ge 156$ an…

Performance of ternary double circulant, double twistulant, and self-dual codes

We study the performance of ternary isodual codes which are not self-dual and ternary self-dual codes, as measured by the decoding error probability in bounded distance decoding. We compare the performance of ternary double circulant and double twistulant codes which are not self-dual with ternary extremal self-dual codes. We also investigate the performance of ternary self-dual codes having large minimum weights.

具有阶数 29 和 31 自同构的 60 阶与 64 阶 Hadamard 矩阵

Hadamard matrices of orders 60 and 64 with automorphisms of orders 29 and 31

本文分类了 p=29、31 时具有 p 阶自同构的 2p+2 阶 Hadamard 矩阵，并据此构造计算，发现了若干新的三元及二元近极值自对偶码。

An Efficient Construction of Self-Dual Codes

We complete the building-up construction for self-dual codes by resolving the open cases over $GF(q)$ with $q \equiv 3 \pmod 4$, and over $\Z_{p^m}$ and Galois rings $\GR(p^m,r)$ with an odd prime $p$ satisfying $p \equiv 3 \pmod 4$ with $r$ odd. We also extend the building-up construction for self-dual codes to finite chain rings. Our building-up construction produces many new interesting self-dual codes. In particular, we construct 945 new extremal self-dual ternary $[32,16,9]$ codes, each of which has a trivial automorphism group. We also obtain many new self-dual codes over $\mathbb Z_9$ o…