Existence of Steiner seven-cycle systems

Abstract

A Steiner k-cycle system of order v is a pair (X,C), where C is a collection of k-cycles of K v based on a v-set X such that for any integer r, 1⩽r⩽k/2 , and for any two distinct vertices x and y of X there exists in C a unique k-cycle along which the distance between x and y is r. Steiner k-cycle systems are useful in constructing authentication perpendicular arrays and authentication and secrecy codes. In this paper, we show that the necessary condition for the existence of Steiner seven-cycle systems, v≡1 or 7 ( mod 14) , is also sufficient if v>861. We also show that there are at most 21 unknown orders below this bound. The result is mainly based on generalized constructions for two holey self-orthogonal Latin squares with symmetric orthogonal mates (2 HSOLSSOM) and some direct constructions. As an application, we shall update the known result on the existence of perfect Mendelsohn designs with block size 7.