Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes with corrigendum

TL;DR

This paper investigates the weight distribution of cyclic codes with an arbitrary number of generalized Niho type zeroes, extending previous results limited to at most three zeroes, and provides explicit weight distributions for these codes.

Contribution

It introduces two new families of cyclic codes with many zeroes and determines their weight distributions, generalizing prior work to arbitrary numbers of zeroes.

Findings

First family has at most (2t+1) non-zero weights.

Second family has at most 2t non-zero weights.

Weight distributions are explicitly determined.

Abstract

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works \cite{Li-Zeng-Hu} and \cite{gegeng2}, we study two families of cyclic codes over $\mathbb{F}_p$ with arbitrary number of zeroes of generalized Niho type, more precisely $\ca$ (for $p=2$) of $t+1$ zeroes, and $\cb$ (for any prime $p$) of $t$ zeroes for any $t$. We find that the first family has at most $(2t+1)$ non-zero weights, and the second has at most $2t$ non-zero weights. Their weight distribution are also determined in the paper.