Searching a $33,16$ Doubly-Even Code for a $22,33,12,8,4$-BIBD

Abstract

It is known that if a $(22,33,12,8,4)$-BIBD exists, then its incidence matrix is contained in a $(33,16)$ doubly-even self-orthogonal code (that does not contain a coordinate of zeros). There are 594 such codes, up to equivalence. It has been theoretically proven that 116 of these codes cannot contain the incidence matrix of such a design. For the remaining 478 codes, an exhaustive clique search may be tried, on the weight 12 words of a code, to determine whether or not it contains such an incidence matrix. Thus far, such a search has been used to show 299 of the 478 remaining codes do not contain the incidence matrix of a $(22,33,12,8,4)$-BIBD.

In this paper, an outline of the method used to search the weight 12 words of these codes is given. The paper also gives estimations on the size of the search space for the remaining 179 codes. Special attention is paid to the toughest cases, namely the 11 codes that contain 0 weight 4 words and the 21 codes that contain one and only one weight 4 word.