[Paper Review] On Repeated-Root Constacyclic Codes of Length $2^amp^r$ over Finite Fields

Aicha Batoul, Kenza Guenda+1 more|arXiv (Cornell University)|May 2, 2015

Coding theory and cryptography16 references3 citations

TL;DR

This paper characterizes the generator polynomials of repeated-root constacyclic codes of length $2^a m p^r$ over $\mathbb{F}_{p^s}$, where $a \geq 1$, $m$ is odd, and $(m,p)=1$. It establishes conditions for the existence of self-dual negacyclic codes and proves that under $q \equiv 1 \pmod{2^{a+1}}$, negacyclic codes of this length are monomially equivalent to cyclic codes, extending prior results on constacyclic codes of composite lengths with repeated roots.

ABSTRACT

In this paper we investigate the structure of repeated root constacyclic codes of length $2^amp^r$ over $\mathbb{F}_{p^s}$ with $a\geq1$ and $(m,p)=1$. We characterize the codes in terms of their generator polynomials. This provides simple conditions on the existence of self-dual negacyclic codes. Further, we gave cases where the constacyclic codes are equivalent to cyclic codes.

Motivation & Objective

• To generalize prior results on constacyclic codes of lengths $2^t p^r$ and $l p^r$ to codes of length $2^a m p^r$ over $\mathbb{F}_{p^s}$ with $a \geq 1$ and $m$ odd and coprime to $p$.
• To characterize the generator polynomials of $\lambda$-constacyclic codes of length $mp^r$ using the structure of codes of length $m$, enabling systematic construction.
• To determine conditions under which repeated-root constacyclic codes are monomially equivalent to cyclic codes, particularly for negacyclic codes.
• To establish necessary and sufficient conditions for the existence of self-dual negacyclic codes of length $2^a m p^r$ over $\mathbb{F}_{p^s}$ when $p$ is odd.
• To extend the theory of self-dual codes to a broader class of repeated-root constacyclic codes, especially in the context of negacyclic structures.

Proposed method

• Uses the ring isomorphism between $\mathbb{F}_{q}[x]/\langle x^n - \lambda \rangle$ and polynomial quotient rings to represent constacyclic codes as ideals.
• Applies $q$-cyclotomic cosets and minimal polynomials over $\mathbb{F}_{q}$ to factor $x^n - \lambda$ and derive generator polynomials.
• Leverages the existence of primitive $2^{a+1}$-th roots of unity in $\mathbb{F}_q^*$ when $q \equiv 1 \pmod{2^{a+1}}$ to construct ring isomorphisms between negacyclic and cyclic code rings.
• Employs factorization of $x^m + 1$ and $x^m - 1$ into irreducible factors over $\mathbb{F}_{p^s}$, grouping self-reciprocal and reciprocal pairs to analyze duality.
• Applies the duality condition $A(x) = B^*(x)$ for negacyclic codes to derive conditions for self-duality via the structure of irreducible factors.
• Uses Lemma 5.6 to link the parity of $\text{ord}_m(p^s)$ to the existence of self-reciprocal minimal polynomials, which determines self-dual code existence.

Experimental results

Research questions

• RQ1 Under what conditions does a self-dual negacyclic code of length $2^a m p^r$ over $\mathbb{F}_{p^s}$ exist when $p$ is odd and $a \geq 1$?
• RQ2 When is a negacyclic code of length $2^a m p^r$ over $\mathbb{F}_{p^s}$ monomially equivalent to a cyclic code?
• RQ3 How can the generator polynomial of a $\lambda$-constacyclic code of length $2^a m p^r$ be explicitly constructed from the generator of a code of length $m$?
• RQ4 What role do $q$-cyclotomic cosets and self-reciprocal irreducible factors play in determining the structure of repeated-root constacyclic codes?
• RQ5 What is the relationship between the order $\text{ord}_m(p^s)$ and the existence of self-dual negacyclic codes in this extended length class?

Key findings

• A negacyclic code of length $2^a m p^r$ over $\mathbb{F}_{p^s}$ is monomially equivalent to a cyclic code if $q \equiv 1 \pmod{2^{a+1}}$, due to the existence of a primitive $2^{a+1}$-th root of unity in $\mathbb{F}_q^*$.
• Self-dual negacyclic codes of length $2^a m p^r$ exist over $\mathbb{F}_{p^s}$ if and only if $\text{ord}_m(p^s)$ is odd, which ensures no self-reciprocal irreducible factors in the factorization of $x^m - 1$.
• The generator polynomial of a negacyclic code of length $2^a m p^r$ is of the form $\prod_{k=1}^{2^a} \left( \prod_{i=0}^{l} f_i^{j_i}(\alpha^{-2k+1}x) \right)$, where $f_i(x)$ are monic irreducible factors of $x^m - 1$ and $\alpha$ is a primitive $2^{a+1}$-th root of unity.
• The existence of self-dual negacyclic codes is equivalent to the absence of self-reciprocal irreducible factors in the factorization of $x^m + 1$, which is determined by the parity of $\text{ord}_m(p^s)$.
• Example: No self-dual negacyclic code of length 70 exists over $\mathbb{F}_5$, but one exists over $\mathbb{F}_9$ for length 126, illustrating the dependence on $\text{ord}_m(p^s)$.