Self-dual 2-quasi Negacyclic Codes over Finite Fields

TL;DR

This work establishes precise existence conditions and asymptotic performance for self-dual $2$-quasi negacyclic codes over finite fields. By introducing the $*$ operator on ${ m R}=F[X]/<X^n+1>$ and leveraging $q$-coset decompositions of $X^n+1$, it derives that self-dual codes exist for all $q$ when $n$ is even, and for odd $n$ precisely when $q ≠ -1 (mod 4)$. It then proves these codes are asymptotically good: for odd $q$ and δ with $h_q(δ)<1/4$, there exists a sequence of codes with length growing to infinity and relative distance exceeding δ, while preserving a rate of $1/2$. The results extend the understanding of GV-bound proximity for self-dual quasi-cyclic families to the negacyclic setting and provide constructive paths for long self-dual codes over finite fields.

Abstract

In this paper, we investigate the existence and asymptotic property of self-dual $2$-quasi negacyclic codes of length $2n$ over a finite field of cardinality $q$. When $n$ is odd, we show that the $q$-ary self-dual $2$-quasi negacyclic codes exist if and only if $q\,{\not\equiv}-\!1~({\rm mod}~4)$. When $n$ is even, we prove that the $q$-ary self-dual $2$-quasi negacyclic codes always exist. By using the technique introduced in this paper, we prove that $q$-ary self-dual $2$-quasi negacyclic codes are asymptotically good.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Example 2.5
• Lemma 2.6
• proof