## The point code of a (22, 33, 12, 8, 4)-balanced incomplete block design

##### Abstract

A ('v, b, r, k', [lambda])-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 [lambda] blocks. The incidence matrix of a ('v, b, r, k', [lambda])-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', [lambda]). 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', [lambda])-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.)