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Quasi-twisted codes and their connection with additive constacyclic codes over finite fields

Kanat Abdukhalikov, Gyanendra K. Verma

Abstract

In this paper, we study quasi-twisted codes and their relationship with additive constacyclic codes through a polynomial-based approach. We first present a polynomial characterization of quasi-twisted codes over finite fields analogous to quasi-cyclic codes and determine Euclidean, Hermitian, and symplectic duals of quasi-twisted codes with index $2$. Additionally, we provide necessary and sufficient conditions for the self-orthogonality of appropriate quasi-twisted codes. Next, we explore a one-to-one correspondence between quasi-twisted codes of length $lm$ with index $l$ over $\mathbb{F}_q$ and additive constacyclic codes of length $m$ over $\mathbb{F}_{q^l}$. We establish relationships between trace inner products in the additive setting and Euclidean, symplectic inner products in the quasi-twisted setting. Using these relations and the correspondence, we determine the dual of additive constacyclic codes with respect to the trace inner products. As a consequence, we conclude that determining the trace Euclidean dual and trace Hermitian dual of an additive constacyclic code is equivalent to determining the Euclidean and symplectic dual of the corresponding quasi-twisted code.

Quasi-twisted codes and their connection with additive constacyclic codes over finite fields

Abstract

In this paper, we study quasi-twisted codes and their relationship with additive constacyclic codes through a polynomial-based approach. We first present a polynomial characterization of quasi-twisted codes over finite fields analogous to quasi-cyclic codes and determine Euclidean, Hermitian, and symplectic duals of quasi-twisted codes with index . Additionally, we provide necessary and sufficient conditions for the self-orthogonality of appropriate quasi-twisted codes. Next, we explore a one-to-one correspondence between quasi-twisted codes of length with index over and additive constacyclic codes of length over . We establish relationships between trace inner products in the additive setting and Euclidean, symplectic inner products in the quasi-twisted setting. Using these relations and the correspondence, we determine the dual of additive constacyclic codes with respect to the trace inner products. As a consequence, we conclude that determining the trace Euclidean dual and trace Hermitian dual of an additive constacyclic code is equivalent to determining the Euclidean and symplectic dual of the corresponding quasi-twisted code.
Paper Structure (10 sections, 42 theorems, 114 equations, 3 tables)

This paper contains 10 sections, 42 theorems, 114 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structure of quasi-twisted codes of index 2
  4. Euclidean dual of quasi-twisted codes
  5. Symplectic dual of quasi-twisted codes
  6. Hermitian dual of quasi-twisted codes
  7. Self-orthogonality of quasi-twisted codes
  8. Quasi-twisted codes over $\mathbb{F}_q$ of index $2$ and additive constacyclic codes over $\mathbb{F}_{q^2}$
  9. Quasi-twisted codes over $\mathbb{F}_q$ of index $l$ and additive constacyclic codes over $\mathbb{F}_{q^l}$
  10. Conclusion

Key Result

Theorem 3.1

1) Let $C$ be a $\lambda$-quasi-twisted code of length $2m$ and index $2$. Then there exist $g_1=(g_{11}(x),g_{12}(x)), g_2=(0,g_{22}(x))\in (\mathbb{F}_q[x])^2$ such that $C$ is generated by $g_1$, $g_2$ and the following holds: Moreover, in this case $\dim C = 2m- \deg g_{11}(x) - \deg g_{22}(x)$. 2) Let code $C$ be generated by elements $g_1=( g_{11}(x),g_{12}(x))$ and $g_2=( 0,g_{22}(x))$, an

Theorems & Definitions (91)

  • Theorem 3.1
  • proof
  • Remark 3.1
  • Theorem 3.2
  • proof
  • Example 3.1
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.1
  • proof
  • ...and 81 more