Hamming code

Algorithm

Pure mathematics

Discrete mathematics

Statistics

Skew

Rank (graph theory)

Telecommunications

Economics

Computer science

Polynomial

Polynomial ring

Hamming space

Decoding methods

Mathematics

Mathematical analysis

Operations management

Reed–Solomon error correction

Block code

Combinatorics

Metric (unit)

Quotient

Algebra over a field

Linear code

Journal of Algebra

We provide an algebraic description for sum-rank metric codes, as quotient space of a skew polynomial ring. This approach generalizes at the same time the skew group algebra setting for rank-metric codes and the polynomial setting for codes in the Hamming metric. This allows to construct twisted linearized Reed-Solomon codes, a new family of maximum sum-rank distance codes extending at the same time Sheekey's twisted Gabidulin codes in the rank metric and twisted Reed-Solomon codes in the Hamming metric. Furthermore, we provide an analogue in the sum-rank metric of Trombetti-Zhou construction, which also provides a family of maximum sum-rank distance codes. As a byproduct, in the extremal case of the Hamming metric, we obtain a new family of additive MDS codes over quadratic fields.

Twisted linearized Reed-Solomon codes: A skew polynomial framework