Trails of triples in partial triple systems

Abstract

Given v, t, and m, does there exist a partial Steiner triple system of order v with t triples whose triples can be ordered so that any m consecutive triples are pairwise disjoint? Given v, t, and m ₁, m ₂, . . . , m _s with ${t = \sum_{i=1}^s m_i}$ , does there exist a partial Steiner triple system with t triples whose triples can be partitioned into partial parallel classes of sizes m ₁, . . . , m _s ? An affirmative answer to the first question gives an affirmative answer to the second when m _i ≤ m for each ${i \in \{1,2,\ldots,s\}}$ . These questions arise in the analysis of erasure codes for disk arrays and that of codes for unipolar communication, respectively. A complete solution for the first problem is given when m is at most ${\frac{1}{3}\left(v-(9v)^{2/3}\right)+{O}\left(v^{1/3}\right)}$ .