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

TL;DR

This paper introduces clique bitrades, a generalization of various known bitrades, and establishes a correspondence with bipartite isometric subgraphs in distance-regular graphs, providing new insights into $q$-ary Steiner trades.

Contribution

It defines clique bitrades, links them to distance-regular graphs, and determines the minimal size of $q$-ary Steiner bitrades, connecting them to dual polar subgraphs.

Findings

Established a one-to-one correspondence between clique bitrades and bipartite isometric subgraphs.

Determined the minimum cardinality of $q$-ary Steiner $T_q(k-1,k,v)$ bitrades.

Connected minimal $q$-ary Steiner bitrades with dual polar subgraphs of Grassmann graphs.

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