Author
SIAP, IRFAN

Year
1999

Advisor
RAY-CHAUDHURI, DIJEN K.

Abstract
MacWilliams-type identities play an important role in coding theory. There has been several generalizations of the Hamming weight enumerator of a code and the identity proved by F. J. MacWilliams in \cite{fj}. We define a generalized $r$-fold weight enumerator of $r$ linear codes over rings, and we obtain a MacWilliams-type identity for the generalized $r$-fold weight enumerators of these codes and their duals. This identity generalizes the previously known MacWilliams-type identities. It gives a unified treatment and combines most well-known MacWilliams-type identities as special cases. By specializing the variables of the generalized $r$-fold weight enumerators we are able to obtain other weight enumerators, such as the Hamming, Lee, complete, $r$-genus, $r$-ply and symmetric weight enumerators. Also, we define generalized $r$-fold symmetric and Hamming weight enumerators of $r$ codes, and MacWilliams-type identities are obtained by specializing the variables in the generalized $r$-fold complete weight enumerators of these codes. Further, we consider codes (submodules) over rings $\mathbb{F}_3 +u\mathbb{F}_3$, $\mathbb{F}_5+u\mathbb{F}_5$ and $\mathbb{F}_5+u\mathbb{F}_5+u^2\mathbb{F}_5$ with $u^2=1$ where $\mathbb{F}_5$ is a finite field with 5 elements. Using a Gray map we relate the codes over these rings with their images which are codes over $\mathbb{F}_3$ and $\mathbb{F}_5$. These "good" submodules give 19 new linear codes over $\mathbb{F}_5$ and one new linear code over $\mathbb{F}_3$ with improved minimum distances. By applying puncturing and shortening to these new codes, one can get further improvements on lower bounds for minimum distances of many other codes. Finally, we apply invariant theory to investigate the ring of invariants of $2$-ply weight enumerators of binary self-dual codes with lengths divisible by 4 or 8. We show that the $2$-ply weight enumerators of binary self dual codes with lengths divisible by 8 are related to the $2$-ply weight enumerators of extended Hamming and Golay codes. We conclude the dissertation by establishing the ring of invariants of complete weight enumerators of codes over $\mathbb{F}_2+u\mathbb{F}_2$ with $u^2=1$ whose images are Type I and II codes.