The complete weight enumerator of the Reed-Solomon code with dimension two or three

TL;DR

This paper explicitly determines the complete weight enumerators of Reed-Solomon and extended Reed-Solomon codes with dimensions 2 and 3 over finite fields, enhancing understanding of their error detection and correction capabilities.

Contribution

It generalizes previous results by explicitly calculating the complete weight enumerators for these codes with specific small dimensions using character sums.

Findings

Complete weight enumerators for k=2,3 Reed-Solomon codes are derived.

Results extend previous work to broader field sizes and code dimensions.

Provides formulas useful for analyzing error detection and correction capabilities.

Abstract

It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are very important for determining the capability of both error-detection and error-correction. In this paper, for any positive integer $m$ and prime $p$, basing on the character sums, we determine the complete weight enumerators of the Reed-Solomon code and the extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over $\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding results in \cite{BK91,K04}.