Low-weight codewords in cyclic codes

TL;DR

This paper presents a new formula for counting low-weight codewords in cyclic codes, linking weight distributions to solutions of polynomial systems, thereby deepening understanding of their structure.

Contribution

It introduces a recursive relationship connecting weight distributions of binary cyclic codes to solutions of polynomial equations, offering a novel analytical approach.

Findings

Derived a formula for weight 2 codewords

Established a recursive relationship for weight distributions

Connected code properties to solutions of polynomial systems

Abstract

We introduce a formula for determining the number of codewords of weight 2 in cyclic codes and provide results related to the count of codewords with weight 3. Additionally, we establish a recursive relationship for binary cyclic codes that connects their weight distribution to the number of solutions of associated systems of polynomial equations. This relationship allows for the computation of weight distributions from known solutions of systems of diagonal equations and vice versa, offering a new insight into the structure and properties of binary cyclic codes.