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

2025-05-09

AI Summary

This paper addresses the efficient decoding of two-dimensional $(lambda_1,lambda_2)$-constacyclic codes over the finite field $mathbb{F}_q$. Confronted with their structural complexity and the high computational cost of conventional decoding methods, we first establish an exact correspondence between two-dimensional spectral zeros and the codeword array structure. We then propose a time-frequency joint sparse modeling framework based on the two-dimensional finite-field Fourier transform (FFFT), enabling precise error-location identification and low-complexity error-value recovery. The method supports codeword arrays of arbitrary dimensions, adjustable parity-area ratios, and flexible code rates. Experimental validation across multiple parameter sets confirms decoding correctness and demonstrates end-to-end error localization and correction. Our approach significantly reduces decoding complexity compared to existing methods, providing both theoretical foundations and practical tools for the real-world deployment of two-dimensional constacyclic codes.

Technology Category

• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Decoding 2-D constacyclic codes over finite fields
• Detecting error locations in 2-D coded arrays
• Exploiting spectral properties for error correction

Innovation

• Uses 2-D FFFT for spectral domain analysis
• Detects 2-D error locations algorithmically
• Decodes errors via time-frequency domain duality