| A (v, b, r, k, λ)-balanced incomplete block design (or simply a BIBD) is a family of b sets, called blocks, each consisting of k elements taken from a set of v elements, called varieties, such that every variety occurs in exactly r blocks and every pair of varieties occur together in exactly λ blocks. The incidence matrix of a (v, b, r, k, λ)-BIBD is a v x b binary matrix A whose rows are indexed by the varieties, typically 1 to v, and whose columns are indexed by the block names, typically 1 to b. Entry ai,j of A contains a 1 if variety i is in block j, otherwise entry ai,j contains a 0.; There are several well-known necessary conditions for the existence of a BIBD with parameters (v, b, r, k, λ). However, these conditions are not sufficient. The parameters with the smallest v that obeys the conditions for which it is not known whether or not a BIBD exists is (22, 33, 12, 8, 4). The problem we will be investigating in this thesis is “does a (22, 33, 12, 8, 4)-BIBD exist?” This has been, and remains, an open problem for over 60 years.; Our approach to this problem is based on the fact that if a (22, 33, 12, 8, 4)-BIBD exists, then so does its point code. The point code of a ( v, b, r, k, λ)-BIBD B is the subspace of Vb(2) that is determined by the span of the rows of the incidence matrix of B. It is known that the point code of a (22, 33, 12, 8, 4)-BIBD is a length 33 doubly-even self-orthogonal code over GF(2).; In this thesis, we will prove that any complete list L of inequivalent (33, 16) doubly-even self-orthogonal codes over GF (2), that do not contain a coordinate of zeros, has the property that a (22, 33, 12, 8, 4)-BIBD exists if and only if L contains a code that contains the incidence matrix of such a design. We have enumerated such a list L of inequivalent (33, 16) doubly-even self-orthogonal codes. We have also found the automorphism group of each code in L. The number of codes in L is 594. (Abstract shortened by UMI.)... |
