Steiner triple systems are well studied combinatorial designs that have been shown to possess properties desirable for the construction of multiple erasure codes in RAID architectures. The ordering of the columns in the parity check matrices of these codes affects system performance. Combinatorial problems involved in the generation of good and bad column orderings are defined, and examined for small numbers of accesses to consecutive data blocks in the disk array. 1 Background A Steiner triple system is an ordered pair (S; T ) where S is a finite set of points or symbols and T is a set of 3-element subsets of S called triples, such that each pair of distinct elements of S occurs together in exactly one triple of T . The order of a Steiner triple system (S; T ) is the size of the set S, denoted jSj. A Steiner triple system of order v is often written as STS(v). An STS(v) exists if and only if v 1; 3 (mod 6) (see [6], for example). We can relax the requirement that every pair ...

Two Steiner triple systems (X,A)and(X,B) are said to intersect in m pairwise disjoint blocks if |A # B| = m and all blocks in A#B are pairwise disjoint. For each v, we completely determine the possible values of m such that there exist two Steiner triple systems of order v intersecting in m pairwise disjoint blocks. 1

The point code of a Steiner triple system always contains vectors of weight equal to the replication number. The existence of Steiner triple systems whose point code has minimum weight less than the replication number is examined. In particular, the possible minimum weights are determined. 1 Background Let V be a finite set of v elements, and let G = fG 1 ; G 2 ; : : : ; G s g be a partition of V into subsets called groups. Let B be a collection of subsets of V called blocks, and set K = fjBj : B 2 Bg, the set of block sizes. If (V; B) has the property that every pair of elements either appears in exactly one block or in exactly one group it is a group divisible design, and is denoted by K-GDD. When K = fkg, the notation k-GDD is typically used. The type of the GDD is z t 1 1 z t 2 2 z t 3 3 \Delta \Delta \Delta z t s s when the number of groups of size z i is t i . A Steiner triple system of order v, or STS(v), is a 3-GDD of type 1 v . Let (V; B) be an STS(v). There are b...

We describe a method used to prove nonexistence of pairwise balanced designs. We determine the exact closure of all subsets K of the set f3; 4; : : : ; 22g with K " f11; 12; : : : ; 22g 6= ; and 3 2 K. 1 Introduction Let K be a set of positive integers. Then a pairwise balanced design (PBD[v; K]) of order v with block sizes from K is a pair (V; B), where V is a finite set (the point set) of cardinality v and B is a family of subsets (called blocks) of V which satisfy the following properties: (i) if B 2 B, then jBj 2 K; (ii) every pair of distinct elements of V occurs in exactly one block of B. A set S of positive integers is said to be PBD-closed if the existence of a PBD[v; S] implies that v belongs to S. Let K be a set of positive integers and let B(K) = fv j 9PBD[v; K]g. Then B(K) is a PBD-closed set called the closure of K. In [4] H.-D.Gronau, R.Mullin and Ch.Pietsch determined the complete closure of all subsets of the set f3; 4; : : : ; 10g which include 3. In this paper...

A 4-cycle system of order n is said to be almost resolvable provided its 4-cycles can be partitioned into (n − 1)/2 almost parallel classes ( = (n − 1)/4 vertex disjoint 4-cycles) and a half parallel class ( = (n−1)/8 vertex disjoint 4-cycles.) We construct an almost resolvable 4-cycle system of every order n ≡ 1 (mod 8) except 9 (for which no such system exists) and possibly 33, 41 and 57. 1

A k-generating set generalizes the notion of a complete arc. Consider the Steiner quasigroup (V; \Phi) of a Steiner triple system. Let W ae V . Then (some) elements of V nW can be written in the form w \Phi b w for w; b w 2 W . In turn, further elements can be written in the form ~ w \Phi (w \Phi b w), and so on. If every element of V can be written using elements of W and at most k \Gamma 1 applications of \Phi, then W is called a k-generating set. When k = 2, such a generating set is a spanning or dominating set, and these have been constructed in the course of constructing STSs with complete arcs. Here we report on the case k = 3, focussing on the situation that every element of V n W can be written in exactly one way as a word with at most two occurrences of \Phi, and using the elements of W . When x = jW j, we find that jV j x + \Gamma x 2 \Delta + x \Delta \Gamma x\Gamma1 2 \Delta for a 3-generating set, and we examine when equality can hold. 1 Definitions A Steine...

The existence of incomplete Steiner triple systems of order v having holes of orders w and u meeting in z elements is examined, with emphasis on the disjoint (z = 0) and intersecting (z = 1) cases. When w u and v = 2w+u \Gamma 2z, the elementary necessary conditions are shown to be sufficient for all values of z. Then for z 2 f0; 1g and v `near' the minimum of 2w + u \Gamma 2z, the conditions are again shown to be sufficient. Consequences for larger orders are also discussed, in particular the proof that when one hole is at least three times as large as the other, the conditions are again sufficient. 1 Introduction Let V be a finite set of v elements, and let G = fG 1 ; G 2 ; : : : ; G s g be a partition of V into subsets called groups. Let B be a collection of subsets of V called blocks, and set K = fjBj : B 2 Bg, the set of block sizes. If (V; B) has the property that every pair of elements either appears in exactly one block or in exactly one group, it is a group divisible de...

Abstract If the complete graph Kn has vertex set X, a maximum packing of Kn with 4-cycles, (X, C, L), is an edge-disjoint decomposition of Kn into a collection C of 4-cycles so that the unused edges (the set L) is as small a set as possible. Maximum packings of Kn with 4-cycles were shown to exist by Schönheim and Bialostocki (Can. Math. Bull. 18:703–708, 1975). An almost parallel class of a maximum packing (X, C, L) of Kn with 4-cycles is a largest possible collection of vertex disjoint 4-cycles (so with ⌊n/4 ⌋ 4-cycles in it). In this paper, for all orders n, except 9, which does not exist, and possibly 23, 41 and 57, we exhibit a maximum packing of Kn with 4-cycles so that the 4-cycles in the packing are resolvable into almost parallel classes, with any remaining 4-cycles being vertex disjoint. [Note: The three missing orders have now been found, and appear in Billington et al. (to appear).]