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Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields

Yun Fan, Yue Leng

Abstract

Let $F$ be a field with cardinality $p^\ell$ and $0\neq λ\in F$, and $0\le h<\ell$. Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois $p^h$-inner product (DCC, vol.84, pp.473-492). In this paper, we characterize the structure of $2$-quasi $λ$-constacyclic codes over $F$; and exhibit necessary and sufficient conditions for $2$-quasi $λ$-constacyclic codes being Galois self-dual. With the help of a technique developed in this paper, we prove that, when $\ell$ is even, the Hermitian self-dual $2$-quasi $λ$-constacyclic codes are asymptotically good if and only if $λ^{1+p^{\ell/2}}=1$. And, when $p^\ell\,{\not\equiv}\,3~({\rm mod}~4)$, the Euclidean self-dual $2$-quasi $λ$-constacyclic codes are asymptotically good if and only if $λ^{2}=1$.

Galois Self-dual 2-quasi Constacyclic Codes over Finite Fields

Abstract

Let be a field with cardinality and , and . Extending Euclidean and Hermitian inner products, Fan and Zhang introduced Galois -inner product (DCC, vol.84, pp.473-492). In this paper, we characterize the structure of -quasi -constacyclic codes over ; and exhibit necessary and sufficient conditions for -quasi -constacyclic codes being Galois self-dual. With the help of a technique developed in this paper, we prove that, when is even, the Hermitian self-dual -quasi -constacyclic codes are asymptotically good if and only if . And, when , the Euclidean self-dual -quasi -constacyclic codes are asymptotically good if and only if .
Paper Structure (12 sections, 33 theorems, 146 equations)

This paper contains 12 sections, 33 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $2$-quasi constacyclic codes over finite fields
  4. Galois self-dual $2$-quasi constacyclic codes
  5. The case that $\lambda^{1+p^h}\neq 1$
  6. The case that $\lambda^{1+p^h}=1$
  7. Hermitian self-dual $2$-quasi-cyclic codes
  8. The operator "$*$" on ${\cal R}=F[X]/\langle X^n-1\rangle$
  9. A class of Hermitian self-dual $2$-quasi-cyclic codes
  10. Hermitian self-dual $2$-quasi-cyclic codes are asymptotically good
  11. Hermitian (Euclidean) self-dual $2$-quasi constacyclic codes
  12. Conclusions

Key Result

Lemma 2.2

(1) Let $C$ be a subspace of $F^n$. Then $C$ is a $\lambda$-constacyclic code if and only if $\mathbf{c}\cdot P_\lambda \in C$, for any $\mathbf{c}\in C$. (2) Let $C$ be a subspace of $F^n\times F^{n}$. Then $C$ is a $2$-quasi $\lambda$-constacyclic code if and only if $(\mathbf{c},\,\mathbf{c}') \

Theorems & Definitions (73)

  • Example 2.1
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Lemma 2.8
  • Remark 3.1
  • Lemma 3.2
  • ...and 63 more