On the existence of certain cyclic difference families and difference matrices

B. T. Rumov

Matematičeskie zametki, Tome 52 (1992) no. 1, pp. 114-119

Résumé

A theorem is proved to the effect that if there exists a $BIB$-schema with parameters $(p^m-1,k,k-1)$, where $k|(p^m-1)$, $p$ is prime, and $m$ is a natural number, then there exists a $BIB$-schema $(p^{mn}-1,k,k-1)$. A consequence is the existnece of a cyclic $BIB$-schema $(p^{mn}-1,p^m-1,p^m-2)$ (($p^m-1$ is prime) that specifies each ordered pair of difference elements at any distance $\rho=1,2,\dots,p^m-2$ (cyclically) precisely once. Recursive theorems on the existence of difference matrices and $(\nu,k,k)$-difference families in the group $Z_v$ of residue classes mod v are proved, along with a theorem on difference families in an additive abelian group.