Given a permutation σ on n symbols {0, 1, . . . , n − 1} and an integer 1 ≤ m ≤ n − 1, the mth contraction of σ is the permutation σ CT m on n − m symbols obtained by deleting the symbols n − 1, n − 2, . . . , n − m from the cycle decomposition of σ. The Hamming distance hd(σ, τ ) between two permutations σ and τ is the number of symbols x such that σ(x) = τ (x), and the Hamming distance of a non-empty set of permutations is the least Hamming distance among all pairs of distinct elements of the set. In this paper we identify how repeated contractions affect the Hamming distance between two permutations, and use it to obtain new lower bounds for the maximum possible size of a set of permutations on q − 1 symbols, for certain prime powers q, having Hamming distance q − 5, and for a set of permutations on q − m symbols, for certain prime powers q and 3 ≤ m ≤ 9, having Hamming distance q − 1 − 2m.
In this paper, we obtain classification results for higher-dimensional analogues of classical association schemes called association schemes on triples (ASTs). We present an algorithm that enumerates all ASTs on a fixed number of vertices whose nontrivial relations are invariant under the action of some group. Applying this algorithm to three, four, and five vertices along with appropriate group actions yields the unique AST over three vertices, the unique symmetric ASTs over four or five vertices, the unique AST over four vertices with two nontrivial relations, and the unique nontrivial circulant AST over five vertices.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.