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 Co-Chair (if applicable)

Biro, Csaba

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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