Two Classes of Reducible Cyclic Codes With Large Minimum Symbol-Pair Distances

Motivated by high-density storage needs, symbol-pair codes were introduced by Cassuto and Blaum to address channels with overlapping symbol outputs. In this paper, we present a systematic study of two families of reducible cyclic codes under the symbol-pair metric. By employing analytical techniques rooted in cyclotomic numbers and Gaussian period theory over finite fields, we characterize the admissible symbol-pair weights of these codes. Significantly, we demonstrate that their minimum symbol-pair distances attain twice the minimum Hamming distances under specific algebraic constraints. Furthermore, we identify and rigorously determine the symbol-pair weight distributions for several three-weight code families. Notably, we construct a class of MDS symbol-pair codes that achieve optimal distance parameters by the puncturing technique. As supplementary contributions, the paper resolves several computational problems concerning generalized cyclotomic numbers, thereby enriching the mathematical foundation for code parameter analysis.