Goppa type sum-rank codes

We construct Goppa type sum-rank codes over \(\mathbb {F}_q\) ${\mathbb{F}}_q$ with the matrix size \(n \times n\) $n\times n$ , directly from Goppa codes or extended Goppa codes over \(\mathbb {F}_{q^n}\) ${\mathbb{F}}_{q^n}$ in the Hamming metric. Lower bounds on dimensions and minimum sum-rank distances of Goppa type sum-rank codes are proved. The Goppa type sum-rank codes offer great flexibility in block length and dimension while maintaining strong error-correcting capability compared to the best known sum-rank codes. Furthermore, we construct numerous distance-optimal binary sum-rank codes with variable block length and minimum sum-rank distance four.