To the theory of $q$-ary Steiner and other-type trades
  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 
Denis Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia),   Ivan Mogilnykh (Sobolev Institute of Mathematics, No...
math.CO