Perfect bitrades in Johnson graphs*

We study perfect and spherical bitrades in Johnson graphs. These combinatorial objects could be used in a switching construction given a 1-perfect code containing it. Despite the fact that the well-known Delsarte hypothesis on nonexistence of perfect codes in Johnson graphs J(n,w) is proven for all Johnson graphs for very wide spectra of n, we show the existence of two nontrivial examples of bitrades arising from the 3 − (12,6,2) Hadamard design and 3-design of the Nordstrom-Robinson code. Using weight distribution approach, we show the nonexistence of perfect bitrades in an infinite series of Johnson graphs.