Optimal cyclic codes with generalized Niho type zeroes and the weight distribution

TL;DR

This paper extends previous work on cyclic codes by computing their weight distribution under relaxed conditions, revealing many are optimal with few non-zero weights through advanced exponential sum analysis.

Contribution

It introduces new methods to determine weight distributions of cyclic codes with generalized Niho type zeroes under less restrictive conditions.

Findings

Many cyclic codes are optimal with few non-zero weights

The weight distribution is explicitly computed for broader code families

Advanced exponential sum techniques are employed

Abstract

In this paper we extend the works \cite{gegeng2,XLZD} further in two directions and compute the weight distribution of these cyclic codes under more relaxed conditions. It is interesting to note that many cyclic codes in the family are optimal and have only a few non-zero weights. Besides using similar ideas from \cite{gegeng2,XLZD}, we carry out some subtle manipulation of certain exponential sums.