Cyclic and Negacyclic Sum-Rank Codes (2401.04885v2)

Abstract: Sum-rank codes have known applications in the multishot network coding, the distributed storage and the construction of space-time codes. U. Mart\'{\i}nez-Pe~{n}as introduced the cyclic-skew-cyclic sum-rank codes and proposed the BCH bound on the cyclic-skew-cyclic sum-rank codes in his paper published in IEEE Trans. Inf. Theory, vol. 67, no. 8, 2021. Afterwards, many sum-rank BCH codes with lower bounds on their dimensions and minimum sum-rank distances were constructed. Sum-rank Hartmann-Tzeng bound and sum-rank Roos bound on cyclic-skew-cyclic codes were proposed and proved by G. N. Alfarano, F. J. Lobillo, A. Neri, and A. Wachter-Zeh in 2022. In this paper, cyclic, negacyclic and constacyclic sum-rank codes are introduced and a direct construction of cyclic, negacyclic and constacyclic sum-rank codes of the matrix size $m \times m$ from cyclic, negacyclic and constacyclic codes over ${\bf F}_{q^m}$ in the Hamming metric is proposed. The cyclic-skew-cylic sum-rank codes are special cyclic sum-rank codes. In addition, BCH and Hartmann-Tzeng bounds for a type of cyclic sum-rank codes are developed. Specific constructions of cyclic, negacyclic and constacyclic sum-rank codes with known dimensions and controllable minimum sum-rank distances are proposed. Moreover, many distance-optimal binary sum-rank codes and an infinite family of distance-optimal binary cyclic sum-rank codes with minimum sum-rank distance four are constructed. This is the first infinite family of distance-optimal sum-rank codes with minimum sum-rank distance four in the literature.