To the theory of $q$-ary Steiner and other-type trades

Abstract

We introduce the concept of a clique bitrade, which generalizes several known types of bitrades, including latin bitrades, Steiner $T(k-1,k,v)$ bitrades, extended $1$-perfect bitrades. For a distance-regular graph, we show a one-to-one correspondence between the clique bitrades that meet the weight-distribution lower bound on the cardinality and the bipartite isometric subgraphs that are distance-regular with certain parameters. As an application of the results, we find the minimum cardinality of $q$-ary Steiner $T_q(k-1,k,v)$ bitrades and show a connection of minimum such bitrades with dual polar subgraphs of the Grassmann graph $J_q(v,k)$. Keywords: bitrades, trades, Steiner systems, subspace designs

13 pp. V2: New Section 2.3, Example 6; now Theorem 4 do not require transitivity. V3: revised, introduction extended

References in corpus (2)

• Large sets of subspace designs
• On the number of maximum independent sets in Doob graphs

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• Combinatorial designs, difference sets and bent functions as perfect colorings of graphs and multigraphs
• A new series of large sets of subspace designs over the binary field
• The minimum volume of subspace trades
• On the cardinality spectrum and the number of latin bitrades of order 3
• $PI$-eigenfunctions of the Star graphs
• On a correspondence between maximal cliques in Paley graphs of square order
• Extremal Peisert-type graphs without the strict-EKR property
• Minimum supports of eigenfunctions of Johnson graphs
• On eigenfunctions of the block graphs of geometric Steiner systems
• MMS-type problems for Johnson scheme
• On extended 1-perfect bitrades