| In chapter 1 we investigate the A(,s)(n,d) problem in the Plotkin Region. The problem is to finding the maximum number of codewords in a code on an alphabet with s symbols that has length n and minimum Hamming distance d. The Plotkin Region is the set of n and d such that sd > t(s - 1)n. We define Generalized Hadamard matrices, using a notion of orthogonality over a group, and show that these matrices give rise to certain A(,s)(n,d) codes. Two general constructions for A(,s)(n,d) codes, which apparently do not depend on Hadamard matrices, are given. Lastly we discuss a computer implemented algorithm to search for A(,3)(15,11).;In chapter 2 we discuss the problem of finding the number of information symbols in a BCH code with design parameters n, b and d. This work generalizes the work of Berlekamp where he completely answers the problem for the case of the "simple" BCH codes (the case b = 0). In the general case (where b is arbitrary) we show the problem to be equivalent to counting certain walks in a directed graph. By considering the adjacency matrix for the graph we find a efficient method of computing I(n,d,b) and we find the asymptotic growth of I(n,d,b) as n, d and b increase while the ratios d/n and b/n are fixed. |
