Binary Cyclic codes with two primitive nonzeros

TL;DR

This paper advances understanding of binary cyclic codes with two primitive nonzeros by exploring their minimum weights and analyzing the Walsh spectrum of specific trace functions over finite fields.

Contribution

It provides new results towards a conjecture on minimum weights and determines the Walsh spectrum for particular trace functions in finite fields.

Findings

Progress on a conjecture regarding minimum weights of certain binary cyclic codes

Explicit determination of Walsh spectrum for specific trace functions

Enhanced understanding of the structure of binary cyclic codes with two primitive nonzeros

Abstract

In this paper, we make some progress towards a well-known conjecture on the minimum weights of binary cyclic codes with two primitive nonzeros. We also determine the Walsh spectrum of $\Tr(x^d)$ over $\F_{2^{m}}$ in the case where $m=2t$, $d=3+2^{t+1}$ and $\gcd(d, 2^{m}-1)=1$.