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

Vidya Sagar, Shikha Patel, Shayan Srinivasa Garani

TL;DR

This work advances the theory and practice of two-dimensional $( ext{lambda}_1, ext{lambda}_2)$-constacyclic codes over finite fields by developing a transform-domain viewpoint using the 2-D finite field Fourier transform. It introduces the CZ/ECZ framework to characterize code structure and its spectral nulls, and derives both time-domain and frequency-domain decoding strategies that exploit sparsity from duality. Three decoding approaches are presented—exhaustive search over error locations, time-domain linear-system solving, and FFT-based spectral decoding—each supported by numerical examples. The results enable flexible code rates and rectangular code areas, with potential impact on 2-D barcodes and high-density storage systems by providing efficient, robust error correction in the transform domain. Overall, the paper bridges algebraic structure with practical decoding techniques in a 2-D setting, expanding the toolkit for multidimensional coding theory.

Abstract

We derive the spectral domain properties of two-dimensional (2-D) $(λ_1, λ_2)$-constacyclic codes over $\mathbb{F}_q$ using the 2-D finite field Fourier transform (FFFT). Based on the spectral nulls of 2-D $(λ_1, λ_2)$-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.

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

TL;DR

This work advances the theory and practice of two-dimensional -constacyclic codes over finite fields by developing a transform-domain viewpoint using the 2-D finite field Fourier transform. It introduces the CZ/ECZ framework to characterize code structure and its spectral nulls, and derives both time-domain and frequency-domain decoding strategies that exploit sparsity from duality. Three decoding approaches are presented—exhaustive search over error locations, time-domain linear-system solving, and FFT-based spectral decoding—each supported by numerical examples. The results enable flexible code rates and rectangular code areas, with potential impact on 2-D barcodes and high-density storage systems by providing efficient, robust error correction in the transform domain. Overall, the paper bridges algebraic structure with practical decoding techniques in a 2-D setting, expanding the toolkit for multidimensional coding theory.

Abstract

We derive the spectral domain properties of two-dimensional (2-D) -constacyclic codes over using the 2-D finite field Fourier transform (FFFT). Based on the spectral nulls of 2-D -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.
Paper Structure (15 sections, 21 theorems, 101 equations, 1 figure, 2 algorithms)

This paper contains 15 sections, 21 theorems, 101 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and background
  4. 1-D Finite Field Fourier Transform and Properties for Constacyclic Codes
  5. Characterization of 2-D Constacyclic Codes in the Transform domain
  6. Two-dimensional finite field Fourier transform
  7. Properties of two-dimensional finite field Fourier transform
  8. Spectral nulls of a 2-D $(\lambda_1, \lambda_2)$-constacyclic code
  9. Decoding Algorithms for 2-D Constacyclic Codes
  10. Error-detections and their locations
  11. Decoding based on exhaustive search over the error locations set: Method I
  12. Decoding in the time-domain: Method II
  13. Decoding using finite field Fourier transforms: Method III
  14. Numerical examples for the above Methods
  15. Conclusions

Key Result

Lemma 1

Let $\lambda \in \mathbb{F}^{\ast}_q$ having order $t_1$ and let $\gamma$ be a primitive $M^{\mathrm{th}}$ root of $\lambda$ in an extension field $\mathbb{F}_{q^t}$ for some positive integer $t$. Let $\alpha$ be a generator of the cyclic group $\mathbb{F}_{q^t}^{\ast}$ then $\gamma=\alpha^{r_1}$, w

Figures (1)

  • Figure 4.1: The parity check locations are organized in contiguous positions in the time domain are hashed and the blank boxes shows the message bit locations. The CZ set identifies the coordinates $(0, 0), (0, 1), (0, 3), (0, 4), (1, 0), (1, 1), (1, 3)$ and $(1, 4)$ as spectral nulls of the given code in Example \ref{['NullExamp']}.

Theorems & Definitions (53)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Example 1
  • Definition 8
  • Remark 1
  • ...and 43 more