Sum-rank product codes and bounds on the minimum distance

Gianira Nicoletta Alfarano, F. Javier Lobillo, Alessandro Neri, and Antonia Wachter-Zeh

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Submission date: 11. Feb. 2022
Bibtex
Keywords and phrases: Sum-rank metric, cyclic codes, skew-cyclic codes, linearized Reed-Solomon codes, tensor product, Hartmann-Tzeng bound, Roos bound
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
The tensor product of one code endowed with the Hamming metric and one endowed with the rank metric is analyzed. This gives a code which naturally inherits the sum-rank metric. Specializing to the product of a cyclic code and a skew-cyclic code, the resulting code turns out to belong to the recently introduced family of cyclic-skew-cyclic. A group theoretical description of these codes is given, after investigating the semilinear isometries in the sum-rank metric. Finally, a generalization of the Roos and the Hartmann-Tzeng bounds for the sum rank-metric is established, as well as a new lower bound on the minimum distance of one of the two codes constituting the product code.