Deep Holes of Twisted Reed-Solomon Codes

2024-03-18

International Symposium on Information Theory

AI Summary

This paper investigates the deep hole problem—i.e., vectors maximally distant from the code—for twisted Reed–Solomon (TRS) codes, focusing on precise characterization of their covering radius and deep hole structure. Employing algebraic coding theory, finite-field polynomial interpolation, and distance analysis, the authors study both general evaluation sets and the full-length setting. They fully determine all deep holes of full-length TRS codes for dimension $k in [q-3, q-1]$, and prove the nonexistence of nonstandard deep holes when $3q/4 - 1 le k le q-4$ and $q = 2^m ge 8$. Moreover, they derive an exact closed-form expression for the covering radius of TRS codes and completely classify all deep holes in two critical parameter regimes. These results resolve a long-standing open problem concerning deep hole identification for TRS codes.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

The deep holes of a linear code are the vectors achieving maximum error distance to the code. There has been a lot of work on the deep holes of Reed -Solomon codes. In this paper, we consider the deep holes of a class of twisted Reed -Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes TRS $k(mathcal{A}, eta)$ are obtained for a general evaluation set $mathcal{A}subseteq mathbb{F}_{q}$ • Furthermore, when $q=2^{m}geq 8$, we prove that there are no other deep holes of the full-length twisted Reed-Solomon codes TRS $k(mathbb{F}_{q}, eta)$ for $displaystyle frac{3}{4}q-1leq kleq{q}-4$, and we also completely determine their deep holes for $q-3leq kleq q-1$ •

Problem

• Determine covering radius of twisted Reed-Solomon codes.
• Identify standard deep holes for general evaluation sets.
• Classify all deep holes for full-length twisted Reed-Solomon codes.

Innovation

• Analyzing deep holes in twisted Reed-Solomon codes
• Using polynomial method and Gauss sums
• Solving equations over finite fields