https://hdl.handle.net/10481/43615 Mon, 06 Apr 2026 07:57:13 GMT 2026-04-06T07:57:13Z https://hdl.handle.net/10481/99242 Roos bound for skew cyclic codes in Hamming and rank metric Alfarano, Gianira; Lobillo Borrero, Francisco Javier; Neri, Alessandro In this paper, a Roos like bound on the minimum distance for skew cyclic codes over a general field is provided. The result holds in the Hamming metric and in the rank metric. The proofs involve arithmetic properties of skew polynomials and an analysis of the rank of parity-check matrices. For the rank metric case, a way to arithmetically construct codes with a prescribed minimum rank distance, using the skew Roos bound, is also given. Moreover, some examples of MDS codes and MRD codes over finite fields are built, using the skew Roos bound. https://hdl.handle.net/10481/99242 https://hdl.handle.net/10481/99231 Sum-rank product codes and bounds on the minimum distance Alfarano, Gianira N.; Lobillo Borrero, Francisco Javier; Neri, Alessandro; Wachter-Zeh, Antonia 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 codes. 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. https://hdl.handle.net/10481/99231 https://hdl.handle.net/10481/99214 Fast parallel computation of reduced row echelon form to find the minimum distance of linear codes Pegalajar Cuéllar, Manuel; Lobillo Borrero, Francisco Javier; Navarro Garulo, Gabriel Finding the distance of linear codes is a key aspect to build error correcting codes, and also to design attacks in code-based post-quantum cryptography; however, it is a NP-hard problem difficult to be addressed. Metaheuristics, and more specifically genetic algorithms, have proven to be a promising tool to improve the search of an upper bound for the distance of a given linear code. In a previous work, it was demonstrated that the there exists a column permutation of a code matrix whose Reduced Row Echelon Form (RREF) contains a row of minimum weight, i.e. the code distance, although calculating RREF during fitness evaluation increases the time complexity of the algorithm substantially. In this work, we propose parallelization of multiple calculations of Reduced Row Echelon Forms simultaneously, and its integration into a fully parallelized design of a CHC evolutionary algorithm to overcome this limitation. Moreover, we demonstrate empirically a substantial improvement in time complexity for the approach in practical case studies to find the distance of linear codes over different finite fields. https://hdl.handle.net/10481/99214 https://hdl.handle.net/10481/99207 Biseparable extensions are not necessarily Frobenius Gómez Torrecillas, José; Lobillo Borrero, Francisco Javier; Navarro Garulo, Gabriel; Sánchez-Hernández, José Patricio We give necessary and sufficient conditions on an Ore extension A[x; σ, δ], where A is a finite dimensional algebra over a field F, for being a Frobenius extension of the ring of commutative polynomials F[x]. As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison. https://hdl.handle.net/10481/99207 https://hdl.handle.net/10481/50292 Global Homological Dimension Of Multifiltered Rings And Quantized Enveloping Algebras Gómez Torrecillas, José; Lobillo Borrero, Francisco Javier https://hdl.handle.net/10481/50292