Date on Master's Thesis/Doctoral Dissertation

8-2022

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Kulosman, Hamid

Committee Member

Biro, Csaba

Committee Member

Gill, Ryan

Committee Member

Li, Jinja

Committee Member

Farag, Aly

Author's Keywords

coding theory; linear complementary dual codes; commutative algebra; cyclic codes; codes over rings

Abstract

A linear code $C$ is called a linear complementary dual code (LCD code) if $C \cap C^\perp = {0}$ holds. LCD codes have many applications in cryptography, communication systems, data storage, and quantum coding theory. In this dissertation we show that a necessary and sufficient condition for a cyclic code $C$ over $\Z_4$ of odd length to be an LCD code is that $C=\big( f(x) \big)$ where $f$ is a self-reciprocal polynomial in $\Z_{4}[X]$ which is also in our paper \cite{GK1}. We then extend this result and provide a necessary and sufficient condition for a cyclic code $C$ of length $N$ over a finite chain ring $R=\big(R,\m=(\gamma),\kappa=R/\m \big)$ with $\nu(\gamma)=2$ to be an LCD code. In \cite{DKOSS} a linear programming bound for LCD codes and the definition for $\text{LD}_{2}(n, k)$ for binary LCD $[n, k]$-codes are provided. Thus, in a different direction, we find the formula for $\text{LD}_{2}(n, 2)$ which appears in \cite{GK2}. In 2020, Pang et al. defined binary $\text{LCD}\; [n,k]$ codes with biggest minimal distance, which meets the Griesmer bound \cite{Pang}. We give a correction to and provide a different proof for \cite[Theorem 4.2]{Pang}, provide a different proof for \cite[Theorem 4.3]{Pang}, examine properties of LCD ternary codes, and extend some results found in \cite{Harada} for any $q$ which is a power of an odd prime.

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