MDS and Self-dual Codes over Rings

In this paper we give the structure of constacyclic codes over formal power series and chain rings. We also present necessary and sufficient conditions on the existence of MDS codes over principal ideal rings. These results allow for the construction of infinite families of MDS self-dual codes over finite chain rings, formal power series and principal ideal rings

Families of Cyclic Codes over Finite Chain Rings

A major difficulty in quantum computation and communication is preventing and correcting errors in the quantum bits. Most of the research in this area has focused on stabilizer codes derived from self-orthogonal cyclic error-correcting codes over finite fields. Our goal is to develop a similar theory for self-orthogonal cyclic codes over the class of finite chain rings which have been proven to also produce stabilizer codes. We also will discuss these restrictions on families of cyclic codes, including, but not limited to quadratic residue codes and Bose-Chaudhuri-Hocquenghem codes. Finally, we will extend the concepts of weight enumerators to the class of Frobenius rings and use them to derive bounds for our codes

Properties and classifications of certain LCD codes.

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