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Title:  The point code of a (22, 33, 12, 8, 4)balanced incomplete block design 
Authors:  Bilous, Richard T. 
Issue Date:  1May2001 
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 wellknown 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 doublyeven selforthogonal code over 'GF'(2). In this thesis, we will prove that any complete list 'L' of inequivalent (33, 16) doublyeven selforthogonal 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) doublyeven selforthogonal 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.) 
URI:  http://hdl.handle.net/1993/2087 
Appears in Collection(s):  FGS  Electronic Theses & Dissertations (Public)

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