A Number Theoretic Conjecture and the Existence of S–Cyclic Steiner Quadruple Systems

• Authors: Helmut Siemon
• Venue: Designs, Codes and Cryptography
• Year: 1998

In their survey article on cyclic Steiner Quadruple Systems SQS(v)M. J. Grannel and T. S. Griggs advanced the conjecture (cf. [8, p. 412])that their necessary condition for the existence of S-cyclic SQS(v) (cf. [7,p. 51]) is also sufficient. Some years prior to that E. Köhler [10]used a graph theoretical method to construct S-cyclic SQS(v). This methodwas extended in [17]-[20] and eventually used to reduce the conjecture ofGrannel and Griggs to a number theoretic claim (cf. also [21], researchproblem 146). The main purpose of the present paper is to attack this claim.For the long intervals we have to distinguish four cases. The proof of casesI–III can be accomplished by a thorough study of how the multiples ofa certain set belonging to the first column of a certain matrix (theelements of which are essentially the vertices of a graph corresponding toSQS(2p)) are distributed over the columns. The proof is by contradiction.Case IV is most difficult to treat and could only be dealt with by very deeplying means. We have to use an asymptotic formula on the number of latticepoints (x,y) with xy ≡ 1 mod p (we speak of 1-points) in a rectangleand this formula shows that the 1-points are equidistributed. But even soour claim could not be proved for all intervals of admissible length.Intervals [a,b] with ^p—_m+1 ^p−_m for some m and 3√4 p^11−15 could not be covered. In thelast section we discuss some conclusions which would follow from thenon-existence of complete intervals.