The Weight Distributions of a Class of Cyclic Codes with Three Nonzeros over F3

TL;DR

This paper investigates the weight distributions of a specific class of ternary cyclic codes with three nonzeros, utilizing higher moments of exponential sums, quadratic forms, and MacWilliams' identities, verified through computational tools.

Contribution

It introduces a novel approach combining higher moments, quadratic forms, and MacWilliams' identities to determine weight distributions of these codes.

Findings

Derived weight distributions for the codes studied.

Validated results with Matlab example.

Highlighted the use of Magma for higher moment calculations.

Abstract

Cyclic codes have efficient encoding and decoding algorithms. The decoding error probability and the undetected error probability are usually bounded by or given from the weight distributions of the codes. Most researches are about the determination of the weight distributions of cyclic codes with few nonzeros, by using quadratic form and exponential sum but limited to low moments. In this paper, we focus on the application of higher moments of the exponential sum to determine the weight distributions of a class of ternary cyclic codes with three nonzeros, combining with not only quadratic form but also MacWilliams' identities. Another application of this paper is to emphasize the computer algebra system Magma for the investigation of the higher moments. In the end, the result is verified by one example using Matlab.