Construction of quasi-cyclic self-dual codes

There is a one-to-one correspondence between $\ell$-quasi-cyclic codes over a finite field $\mathbb F_q$ and linear codes over a ring $R = \mathbb F_q[Y]/(Y^m-1)$. Using this correspondence, we prove that every $\ell$-quasi-cyclic self-dual code of length $m\ell$ over a finite field $\mathbb F_q$ can be obtained by the {\it building-up} construction, provided that char $(\mathbb F_q)=2$ or $q \equiv 1 \pmod 4$, $m$ is a prime $p$, and $q$ is a primitive element of $\mathbb F_p$. We determine possible weight enumerators of a binary $\ell$-quasi-cyclic self-dual code of length $p\ell$ (with $p$ a prime) in terms of divisibility by $p$. We improve the result of [3] by constructing new binary cubic (i.e., $\ell$-quasi-cyclic codes of length $3\ell$) optimal self-dual codes of lengths $30, 36, 42, 48$ (Type I), 54 and 66. We also find quasi-cyclic optimal self-dual codes of lengths 40, 50, and 60. When $m=5$, we obtain a new 8-quasi-cyclic self-dual $[40, 20, 12]$ code over $\mathbb F_3$ and a new 6-quasi-cyclic self-dual $[30, 15, 10]$ code over $\mathbb F_4$. When $m=7$, we find a new 4-quasi-cyclic self-dual $[28, 14, 9]$ code over $\mathbb F_4$ and a new 6-quasi-cyclic self-dual $[42,21,12]$ code over $\mathbb F_4$.Comment: 25 pages, 2 tables; Finite Fields and Their Applications, 201

Additive Self-Dual Codes over GF(4) with Minimal Shadow

We define additive self-dual codes over G F ( 4 ) with minimal shadow, and we prove the nonexistence of extremal Type I additive self-dual codes over G F ( 4 ) with minimal shadow for some parameters

Additive Self-Dual Codes over GF(4) with Minimal Shadow

We define additive self-dual codes over G F ( 4 ) with minimal shadow, and we prove the nonexistence of extremal Type I additive self-dual codes over G F ( 4 ) with minimal shadow for some parameters

MDS Self-Dual Codes and Antiorthogonal Matrices over Galois Rings

In this study, we explore maximum distance separable (MDS) self-dual codes over Galois rings G R ( p m , r ) with p ≡ − 1 ( mod 4 ) and odd r. Using the building-up construction, we construct MDS self-dual codes of length four and eight over G R ( p m , 3 ) with ( p = 3 and m = 2 , 3 , 4 , 5 , 6 ), ( p = 7 and m = 2 , 3 ), ( p = 11 and m = 2 ), ( p = 19 and m = 2 ), ( p = 23 and m = 2 ), and ( p = 31 and m = 2 ). In the building-up construction, it is important to determine the existence of a square matrix U such that U U T = − I , which is called an antiorthogonal matrix. We prove that there is no 2 × 2 antiorthogonal matrix over G R ( 2 m , r ) with m ≥ 2 and odd r

On the Problem of the Existence of a Square Matrix U Such That UUT = −I over Zpm

Building-up construction is one of several methods for constructing self-dual codes. Recently, a new building-up construction method has been developed by S. Han, in which the existence of a square matrix U such that U U T = - I is essential. In this paper, we completely solve the existence problem for U over Z p m , where p is an arbitrary prime number

On the Problem of the Existence of a Square Matrix U Such That UUT = −I over Zpm

Building-up construction is one of several methods for constructing self-dual codes. Recently, a new building-up construction method has been developed by S. Han, in which the existence of a square matrix U such that U U T = - I is essential. In this paper, we completely solve the existence problem for U over Z p m , where p is an arbitrary prime number

Near-Extremal Type I Self-Dual Codes with Minimal Shadow over GF(2) and GF(4)

Binary self-dual codes and additive self-dual codes over GF(4) contain common points. Both have Type I codes and Type II codes, as well as shadow codes. In this paper, we provide a comprehensive description of extremal and near-extremal Type I codes over GF(2) and GF(4) with minimal shadow. In particular, we prove that there is no near-extremal Type I [24m,12m,2m+2] binary self-dual code with minimal shadow if m≥323, and we prove that there is no near-extremal Type I (6m+1,26m+1,2m+1) additive self-dual code over GF(4) with minimal shadow if m≥22