Cyclic codes as submodules of rings and direct product of rings /

• Autores: Roger Ten Valls
• Directores de la Tesis: Cristina Fernández Córdoba (dir. tes.) , Joaquim Borges Ayats (dir. tes.)
• Lectura: En la Universitat Autònoma de Barcelona ( España ) en 2017
• Idioma: español
• Tribunal Calificador de la Tesis: Mercè Villanueva Gay (presid.) , José Joaquín Bernal Buitrago (secret.) , Sergio R. López Permouth (voc.)
• Enlaces
  • Tesis en acceso abierto en:  DDD  TDX 
• Resumen

  • Cyclic codes are an important family in coding theory and have been a primary area of study since its inception. Until the 1990s the usual alphabet chosen by coding theorist was a finite field. Thereafter, it began the study of codes over rings.

    Since the emergence of Z2Z4-additive codes, the research on codes over mixed ring alphabets has increased. In 2014, Abualrub et al. presented Z2Z4-additive cyclic codes and it marked the beginning of the study of cyclic properties on codes over mixed alphabets.

    This thesis aims to explore the algebraic structure of cyclic codes as submodules of direct product of finite rings. As these codes can be seen as submodules of the direct product of polynomial rings, we determine the structure of these codes giving their generator polynomials. Further, we study the concept of duality defining the corresponding polynomial operation to the inner product of vectors. This operation allows us to understand the duality in the corresponding polynomial ring. Moreover, we provide techniques to give a polynomial description for dual codes in terms of the generator polynomials of the cyclic codes and we compute them in some particular cases.

    Also, we consider different metrics in the direct product of finite rings and we study their binary images under distinct distance preserving maps, called Gray maps.

    Finally, we give an algebraic structure for a large family of binary quasi-cyclic codes constructing a family of commutative rings and a canonical Gray map, such that cyclic codes over this family of rings produce quasi-cyclic codes of arbitrary index in the Hamming space via the Gray map.