Optimal and Pessimal Orderings of Steiner Triple Systems in Disk Arrays

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 occurs exactly once as follows. Let (V;B) be a set V of elements together with a collection B of 3-element subsets of V , so that no pair of elements of V occurs as a subset of more than one B 2 B. Such a pair (V;B) is an (n; `)-configuration when n = jV j and ` = jBj, and every element of V is in at least one of the sets in B. Let C be a configuration (V;B). We examine the following combinatorial problem. When does there exist a Steiner triple system (S; T ) of order v in which the triples can be ordered T0; : : : ; Tb 1, so that every ` consecutive triples form a configuration isomorphic to C? Such an ordering is a Cordering of the Steiner triple system. When we treat the first triple as following the last (and hence cyclically order the triples), and then enforce the same condition, the ordering is a C-cyclic ordering. The presence of configurations in Steiner triple systems has been studied in much detail;