On constacyclic codes and their generalizations

• Main Author: Tharnnukhroh, Jareena
• Other Authors: Ling San
• Format: Thesis-Doctor of Philosophy
• Language: English
• Published: Nanyang Technological University 2022
• Subjects: Science::Mathematics::Algebra; Science::Mathematics::Applied mathematics::Information theory
• Online Access: https://hdl.handle.net/10356/161191
• Institution: Nanyang Technological University

Algebraic codes are of interest due to their rich algebraic structures and links with other mathematical objects. Some algebraic codes also have good parameters, while some have found applications. In this thesis, three families of algebraic codes over finite fields are studied, namely, Type-II polyadic constacyclic codes, quasi-twisted codes and generalized negacyclic codes. The results are summarized as follows. For a family of Type-II polyadic constacyclic codes, the existence of such codes is determined using the length of orbits under a suitable group action. A necessary condition and a sufficient condition for a positive integer s to be a multiplier of a Type-II m-adic constacyclic code are determined. Subsequently, for a given positive integer m, a necessary condition and a sufficient condition for the existence of Type-II m-adic constacyclic codes are derived. In many cases, these conditions become both necessary and sufficient. For the other cases, determining necessary and sufficient conditions is equivalent to the discrete logarithm problem which is considered to be computationally intractable. Some special cases are investigated together with examples of Type-II polyadic constacyclic codes with good parameters. For a family of quasi-twisted codes, spectral bounds on their minimum distances are given using eigenvalues of polynomial matrices and the corresponding eigenspaces. These bounds generalize the Semenov-Trifonov and Zeh-Ling bounds in a way analogous to how the Roos and shift bounds extend the BCH and Hartmann{Tzeng (HT) bounds for cyclic codes. The eigencodes of a quasi-twisted code in the spectral theory and the outer codes in its concatenated structure are related. A comparison based on this relation verifies that the Jensen bound always outperforms the spectral bound under special conditions, which yields a similar relation between the Lally and the spectral bounds. The performances of the Lally, Jensen and spectral bounds are given in comparison with each other. For a family of generalized negacyclic codes, the algebraic structure of such codes is established through cyclotomic classes in abelian groups and ideals in twisted group algebras. Recursive constructions and enumerations of such codes are presented. Characterizations of self-dual generalized negacyclic codes and complementary dual generalized negacyclic codes are given as well as their enumerations.