On a generalization of both affine planes and MDS-codes with applications to experiment planning
In 1986, \textit{W. Benz} [Ann. Discrete Math. 30, 15-30 (1986; Zbl 0591.51017)] introduced the very general concept of a \(T(t,q,r,n)\) structure, which, by various choices of the parameters, may be a 3-net, an affine plane, a Laguerre plane, a Minkowski plane, an MDS-code or certain generalizations of these structures. In 1992, \textit{W. Benz} [Ann. Discrete Math. 52, 25-36 (1992; Zbl 0768.51012)] reformulated the definition, basing it on the concept of a \((q,r,n)\)-matrix. Given a set \(P\), with \(|P|= qr\), a \((q,r,n)\)-matrix is an \(n \times r\) matrix whose entries are \(q\)-subsets of \(P\), such that each row is a partition of \(P\) and the cardinality of the intersection of any \(j\) \( (1 \leq j \leq n )\) entries, with no two entries selected from the same row, is independent of the choice of entries. In the original formulation, \(n\) \(q \times r\) matrices, whose entries are elements of \(P\), were used. Each of these, upon amalgamating the columns into sets, becomes a row of a \((q,r,n)\)-matrix in the new formulation. The author of the work under review provides a simplification of the definition of a \((q,r,n)\)-matrix by showing that in the definition it suffices to consider only sets of \(n\) entries, no two from the same row. He then proceeds to taking the amalgamation process one step further, removing the matrix structure from a \((q,r,n)\)-matrix (and hence, the implicit orderings imposed by that structure) by defining a \((q,r,n)-set\) as a set of \(n\) partitions of \(P\) with the appropriate properties. This leads to no additional clarity or applicability of the underlying concept. An \(r\)-set \(X\) of \(P\) is called an interaction of a \((q,r,n)\)-matrix (or set) if every component of every partition meets \(X\) (in a unique point). Let \(M\) be a \((q,r,n)\)-matrix and \(B\) a set of interactions of \(M\). For \(t \leq r\), \((M,B)\) is called a \(T(t,q,r,n)\)-structure if, and only if, for every \(t\)-subset \(\Delta\) of \(P\) with no two elements of \(\Delta\) in the same component of any partition of \(M\), there is exactly one interaction \(X \in B\) with \(\Delta \subset X\). The author examines isomorphisms between \(T(t,q,r,n)\)-structures and the equivalence of certain \(T\)-structures and sharply \(t\)-transitive sets of permutations of degree \(r\). These ideas are exploited in the 2-transitive case, where such sets can be obtained from ternary rings, and, as a special case, nearfields. The \(T(2,r^{n- 1},r,n)\)-structures arising from ternary rings can be characterized with the use of the lattice-theoretic \textit{diagonal axioms}. The characterization of these structures arising from nearfields also utilizes the lattice-theoretic \textit{rectangle axioms}. These lattice-theoretic concepts, first appearing in the author's Diplomarbeit and Dissertation, are reformulated here to provide a more satisfactory characterization.
scientific article; zbMATH DE number 1053895
scientific article