核心概念
Twisted Reed-Solomon codes' deep holes are explored, determining covering radius and standard deep holes.
摘要
The article delves into the investigation of deep holes in twisted Reed-Solomon codes. It covers the abstract concept of deep holes in linear codes, focusing on TRS codes. The study aims to determine the covering radius and a standard class of deep holes for a specific evaluation set. Various theorems and results are presented to establish conditions for vectors to be considered as deep holes in TRS codes. The content is structured into sections discussing definitions, results, organization, preliminaries, determinations of covering radius and deep holes, and conclusions.
Definitions:
- Introduction to linear codes and Hamming distance.
- Definition of Reed-Solomon codes RSk(A).
Results:
- Determination of covering radius ρ(C) in linear codes.
- Extensive research on deep holes in Reed-Solomon codes.
Organization:
- Presentation of results on determinants and character sums.
- Determination of covering radius and standard deep holes in twisted RS codes.
- Results on completeness of deep holes in full-length twisted RS codes.
Preliminaries:
- Notations used throughout the paper explained.
Determinations:
- Calculation of determinants for analysis purposes.
- Propositions regarding group characters and exponential sums discussed.
Conclusions:
- Summary of findings regarding the completeness of deep holes in TRS codes based on even or odd q values.
統計資料
McLoughlin [1] has proven that determining the covering radius exceeds NP-completeness difficulty.
Li and Wan [6] proved that vectors with generating polynomials determine non-deep holes for certain cases.