Constructions of Abelian Codes multiplying dimension of cyclic codes

TL;DR

This work develops a method to lift univariate cyclic (and Reed-Solomon) codes to bivariate abelian codes by exploiting the $q$-orbit structure and the strong apparent distance within a multivariate BCH framework. It multiplies code dimension by a factor $n$ while preserving the true minimum distance in many cases, notably when cyclic codes satisfy $d_{\max BCH}=d$. The authors show that, for cyclic codes with $sd^*(C)=\delta$, the corresponding abelian codes $C_n$ satisfy $sd^*(C_n)=\delta$ and $d(C_n)=d(C)$, with $\dim(C_n)=n\,\dim(C)$, and they provide explicit constructions extending BCH and Reed-Solomon codes to multivariate abelian codes. The RS-enabled examples illustrate substantial dimension growth without sacrificing minimum distance, and the results highlight the non-MDS nature of these codes along with potential decoding implications.

Abstract

In this note, we apply some techniques developed in [1]-[3] to give a particular construction of bivariate Abelian Codes from cyclic codes, multiplying their dimension and preserving their apparent distance. We show that, in the case of cyclic codes whose maximum BCH bound equals its minimum distance the obtained abelian code verifies the same property; that is, the strong apparent distance and the minimum distance coincide. We finally use this construction to multiply Reed-Solomon codes to abelian codes

Theorems & Definitions (23)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Theorem 3.4
• Definition 3.5
• Proposition 3.6
• Definition 3.7
• Example 3.8
• Lemma 4.1
• proof