Weight Distribution of Repeated-Root Cyclic Codes with Prime Power Lengths

Determining the weight distribution of a linear code is a classical and fundamental topic in coding theory that has been extensively investigated. Repeated-root cyclic codes, which form a significant subclass of error-correcting codes, have found broad applications in quantum error-correcting codes, symbol-pair codes, and storage codes. Through polynomial derivation, we derive the monomial equivalent codes for these repeated-root cyclic codes with prime power lengths. Given that monomial equivalent codes exhibit identical weight distributions, we transform the computation of the weight distribution of these repeated-root cyclic codes into the computation of the weight distribution of their monomial equivalent codes. Leveraging the classical results on the weight distribution of MDS codes, we explicitly determine the weight distribution of these repeated-root cyclic codes. Moreover, we apply the weight distribution formula to construct a class of $p$-weight cyclic codes for any prime $p$.