Decoding Algorithms for Two-dimensional Constacyclic Codes over $\mathbb{F}_q$

TL;DR

This paper develops spectral domain analysis and decoding algorithms for two-dimensional constacyclic codes over finite fields, enabling efficient error detection and correction with flexible code parameters.

Contribution

It introduces a spectral domain characterization of 2-D constacyclic codes and proposes novel decoding algorithms leveraging time-frequency domain duality and sparsity.

Findings

Effective error detection algorithm demonstrated.

Decoding algorithms successfully extract error values.

Flexible code construction applicable to various code areas.

Abstract

We derive the spectral domain properties of two-dimensional (2-D) $(\lambda_1, \lambda_2)$-constacyclic codes over $\mathbb{F}_q$ using the 2-D finite field Fourier transform (FFFT). Based on the spectral nulls of 2-D $(\lambda_1, \lambda_2)$-constacyclic codes, we characterize the structure of 2-D constacyclic coded arrays. The proposed 2-D construction has flexible code rates and works for any code areas, be it odd or even area. We present an algorithm to detect the location of 2-D errors. Further, we also propose decoding algorithms for extracting the error values using both time and frequency domain properties by exploiting the sparsity that arises due to duality in the time and frequency domains. Through several illustrative examples, we demonstrate the working of the proposed decoding algorithms.