Optimal three-weight cyclic codes whose duals are also optimal

TL;DR

This paper introduces a new class of optimal three-weight cyclic codes of length q+1 and dimension 3 over finite fields, and demonstrates that their duals are also optimal with known weight distributions, expanding the understanding of cyclic code structures.

Contribution

The paper presents a novel class of optimal three-weight cyclic codes with specific parameters and analyzes their duals, including weight distribution using Krawtchouck polynomials.

Findings

New class of optimal three-weight cyclic codes with weights q-1, q, q+1

Dual codes are also optimal cyclic codes with minimum Hamming distance 4

Weight distribution of dual codes obtained via Krawtchouck polynomials

Abstract

A class of optimal three-weight cyclic codes of dimension 3 over any finite field was presented by Vega [Finite Fields Appl., 42 (2016) 23-38]. Shortly thereafter, Heng and Yue [IEEE Trans. Inf. Theory, 62(8) (2016) 4501-4513] generalized this result by presenting several classes of cyclic codes with either optimal three weights or a few weights. Here we present a new class of optimal three-weight cyclic codes of length $q+1$ and dimension 3 over any finite field $F_q$, and show that the nonzero weights are $q-1$, $q$, and $q+1$. We then study the dual codes in this new class, and show that they are also optimal cyclic codes of length $q+1$, dimension $q-2$, and minimum Hamming distance $4$. Lastly, as an application of the Krawtchouck polynomials, we obtain the weight distribution of the dual codes.