Recent Advances in Difference Sets

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