Date

2012

Author

Özadam, Hakan

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We study the Hamming distance and the structure of repeated-root cyclic codes, and their generalizations to constacyclic and polycyclic codes, over finite fields and Galois rings. We develop a method to compute the Hamming distance of these codes. Our computation gives the Hamming distance of constacyclic codes of length $np^s$\ in many cases. In particular, we determine the Hamming distance of all constacyclic, and therefore cyclic and negacyclic, codes of lengths p^s and 2p^s over a finite field of characteristic $p$. It turns out that the generating sets for the ambient space obtained by torsional degrees and strong Groebner basis for the ambient space are essentially the same and one can be obtained from the other. In the second part of the thesis, we study matrix product codes. We show that using nested constituent codes and a non-constant matrix in the construction of matrix product codes with polynomial units is a crucial part of the construction. We prove a lower bound on the Hamming distance of matrix product codes with polynomial units when the constituent codes are nested. This generalizes the technique used to construct the record-breaking examples of Hernando and Ruano. Contrary to a similar construction previously introduced, this bound is not sharp and need not hold when the constituent codes are not nested. We give a comparison of this construction with a previous one. We also construct new binary codes having the same parameters, of the examples of Hernando and Ruano, but non-equivalent to them.

Subject Keywords

Coding theory., Computer security., Code generators., Galois theory.

URI

http://etd.lib.metu.edu.tr/upload/12615304/index.pdf
https://hdl.handle.net/11511/22177

Collections

Graduate School of Applied Mathematics, Thesis