Recent Advances in Difference Sets

Abstract

According to Schuetzenberger [5] we also know, that if v is even then k-X is a square. The incidence matrix of a (bib) with parameters (4N1, 2N -1, N-1) can be used to construct Hadamard matrices of order 4N. To do this one replaces the zeros of the incidence matrix by -1 and borders the resulting matrix by a row and a column of l's. Other Hadamard matrices may be obtained directly (without bordering) from (bib) designs with parameters 4N2, 2N2 N, N2 N. Hadamard matrices have been used in the construction of binary codes [4], and there is no reason why other (bib) designs should not prove advantageous especially in asymmetric channels. It seems also reasonable to expect that codes constructed from groups and especially from Abelian groups will be relatively easy to implement. Apart from its intrinsic interest as a problem in combinatorial analysis, therefore, the construction of difference sets and the question of their existence for certain parameter combinations is of interest in the theory of error correcting codes. We shall restrict ourselves here principally to Abelian groups because not much is known about difference sets in non-Abelian groups. The difference sets with k =v and with k = v-I are called trivial and will not be considered here. It is easy to see that the complement of every difference set is a difference set so that we may always assume k <v/2. The research here goes in two directions: The construction of new sets, and the proof of the impossibility of solutions for certain parameter combinations. While progress in construction of difference sets was comparatively slow, an impressive number of theorems ruling out many infinite classes of parameter combinations have been obtained in recent years. These impossibility proofs sometimes refer to all Abelian groups and sometimes to specific groups only. Cyclic and elementary Abelian groups have received special attention. A complete result has been obtained for groups of order 2m. It has been shown [3] that the only possible solutions for v, k, X are v=22m, k =22m-1_2m-1, X=22m-2_2m-1. Difference sets for these parameter combinations can be constructed easily in groups which are the direct products of fourgroups and of cyclic groups of order 4. According to R. Turyn [7] one can take