Analytical determination of weight structures in irreducible cyclic codes via Gaussian periods

This paper investigates the weight structure of irreducible cyclic codes, which plays a central role in understanding their error-correcting capabilities. Existing studies typically analyse only specific subclasses of these codes, such as semi primitive or Paley-type cases, leading to fragmented results. To address this limitation, we present a unified approach based on Gaussian periods for analysing irreducible cyclic codes in a general setting. Using this framework, we show that the weight of each codeword depends solely on its associated cyclotomic class, which allows all nonzero Hamming weights to be expressed in compact closed forms and enables the systematic determination of key code parameters, including the weight distribution and minimum distance. The same approach is further extended to the b-symbol metric, demonstrating that both single-symbol and multi-symbol weight structures can be treated within a common analytical framework. The proposed method applies uniformly for all admissible choices of q,r, and N, without imposing restrictive algebraic conditions. Numerical examples are provided to illustrate the theory and to verify the accuracy of the derived expressions. Overall, the results offer a clear and unified Gaussian-period-based methodology for evaluating the fundamental parameters of irreducible cyclic codes.