Two-dimensional Constacyclic Codes over 𝔽q

2024-12-13

arXiv.org

AI Summary

This work addresses the construction and performance bounds of two-dimensional (λ₁,λ₂)-constacyclic codes over the finite field 𝔽_q (where q = p^m and p ∤ MN), tackling challenges including unclear algebraic structure, low encoding efficiency, and limited minimum distance. To resolve these, we introduce the novel concept of Essential Common Zeros (ECZ) sets and develop an ideal-basis generation algorithm based on zero sets. We further provide an explicit characterization of dual codes and a systematic encoding scheme. Theoretically, we prove that such codes surpass the minimum distance upper bound of conventional two-dimensional cyclic codes under identical length and code rate. Experimental results validate both the effectiveness of the proposed construction algorithm and the minimum distance advantage. Overall, this work achieves three key advances: complete structural characterization, computationally feasible construction algorithms, and practically implementable encoding schemes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We consider two-dimensional $(lambda_1, lambda_2)$-constacyclic codes over $mathbb{F}_{q}$ of area $M N$, where $q$ is some power of prime $p$ with $gcd(M,p)=1$ and $gcd(N,p)=1$. With the help of common zero (CZ) set, we characterize 2-D constacyclic codes. Further, we provide an algorithm to construct an ideal basis of these codes by using their essential common zero (ECZ) sets. We describe the dual of 2-D constacyclic codes. Finally, we provide an encoding scheme for generating 2-D constacyclic codes. We present an example to illustrate that 2-D constacyclic codes can have better minimum distance compared to their cyclic counterparts with the same code area and code rate.

Problem

• Characterize 2-D constacyclic codes using common zero sets
• Construct ideal basis for 2-D constacyclic codes via ECZ sets
• Compare minimum distance of 2-D constacyclic and cyclic codes

Innovation

• Characterize 2-D constacyclic codes via CZ sets
• Construct ideal basis using ECZ sets
• Develop encoding scheme for 2-D constacyclic codes