On constacyclic codes and their generalizations
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 p...
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| Format: | Thesis-Doctor of Philosophy |
| Language: | English |
| Published: |
Nanyang Technological University
2022
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| Online Access: | https://hdl.handle.net/10356/161191 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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. |
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