Binary codes from graphs related to the Johnson graph Γ(n, k, k − 1)

Abstract

We show that the binary codes generated by the row span of adjacency matrices of the uniform subset graph Γ(2k, k, 1) and the Johnson graph Γ(2k, k, k − 1) coincide despite the graphs being non-isomorphic. We extend our results to the binary codes of Γ(2k, k, i) and k Γ(2k, k, k − i) where (k even), by showing that their adjacency matrices are equivalent. Further, we discuss the binary codes from the generalised uniform subset graph Γ(2k, k, I) for I = {1, k − 1}, and show that they are self-orthogonal for k ≥ 3.