Low-weight codewords in cyclic codes

The article explores the computation of low-weight codewords in cyclic codes, a critical parameter in error-correcting codes. It delves into the properties of cyclic codes, including their definition, weight distribution, and minimum distance. The study focuses on binary cyclic codes, presenting a recursive relation between the weight distribution and the number of solutions of suitable systems of diagonal equations. The main results are applied to determine the weight distribution of specific families of cyclic codes and to compute the number of solutions of systems of diagonal equations. The article concludes with practical applications and examples, showcasing the significance of low-weight codewords in enhancing the efficiency and reliability of error-correcting codes.