Codis hadamard, quasi-hadamard i generalitzats hadamard full propelinear

• Autores: Iván Bailera Martín
• Directores de la Tesis: Joaquim Borges Ayats (dir. tes.)
• Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2020
• Idioma: español
• ISBN: 9788449094781
• Tribunal Calificador de la Tesis: Mercè Villanueva Gay (presid.), Raúl M. Falcón (secret.), D. L. Flannery (voc.)
• Programa de doctorado: Programa de Doctorado en Informática por la Universidad Autónoma de Barcelona
• Enlaces
  • Tesis en acceso abierto en: TDX
• Resumen

  • This thesis belongs to the fields of algebraic combinatorics and mathematical information theory. Motivated by the computational advantage of the full propelinear structure, we study different kinds of error-correcting codes endowed with this structure. Since a full propelinear code is also a group, it is possible to generate the code from the codewords associated to the generators as a group, even if the code is nonlinear. This offers the data storage benefits of a linear code. Rifà and Suárez introduced full propelinear codes based on binary Hadamard matrices (HFP-codes) and they proved an equivalence with Hadamard groups. The existence of Hadamard matrices of orders a multiple of four remains an open problem. Therefore, the study of new Hadamard codes may contribute to address the Hadamard conjecture. A code with a full propelinear structure is composed of two sets, i.e., codewords and permutations. We define the associated group of an HFP-code as the group comprised of the permutations. Firstly, we study the HFP-codes with a fixed associated group. The next step is to generalize the binary HFP-codes to finite fields. Subsequently, we prove that the existence of generalized Hadamard full propelinear codes is equivalent to the existence of central relative (v,w,v,v/w)-difference sets. Furthermore, we build infinite families of nonlinear generalized Hadamard full propelinear codes. Finally, we introduce the concept of quasi-Hadamard full propelinear code. We also give an equivalence between quasi-Hadamard groups and quasi-Hadamard full propelinear codes. In all codes studied, we analyze the rank and the dimension of the kernel. Two parameters that provide information about the linearity of a code, and also about the nonequivalence of codes.