The weight distribution of a family of p-ary cyclic codes

TL;DR

This paper determines the weight distribution of a specific family of p-ary cyclic codes with duals having two zeros, expanding understanding of their structure and properties in coding theory.

Contribution

It explicitly computes the weight distribution of a new family of cyclic codes with particular dual zero conditions, which was previously unknown.

Findings

Derived explicit weight distributions for the codes.

Identified conditions on t for the code properties.

Enhanced understanding of code structure and dual zeros.

Abstract

Let m, k be positive integers, p be an odd prime and $\pi $ be a primitive element of $\mathbb{F}_{p^m}$. In this paper, we determine the weight distribution of a family of cyclic codes $\mathcal{C}_t$ over $\mathbb{F}_p$, whose duals have two zeros $\pi^{-t}$ and $-\pi^{-t}$, where $t$ satisfies $t\equiv \frac{p^k+1}{2}p^\tau \ ({\rm mod}\ \frac{p^m-1}{2}) $ for some $\tau \in \{0,1,\cdots, m-1\}$.