Classification of tight symmetric inner distribution codes in Johnson schemes

Determine whether every tight symmetric inner distribution code for the Johnson scheme J(2n,n) has degree s=1 or s=n-1.

Background

The paper studies symmetric inner distribution codes in the Johnson association scheme J(2n,n), equivalently constant-weight binary codes of length 2n and weight n whose distance sets are symmetric about n/2. The authors prove the upper bound |C|≤\binom{2n-1}{s} for a code of degree s and analyze the equality case, calling such codes tight.

For a tight code, the authors establish that an associated polynomial \Phi_s(z) must have distinct integral zeros. They report that computer experiments indicate that these zeros are rarely all integers except when s=1 or r=1, where n=r+s; the condition r=1 is equivalent to s=n-1. The conjecture asserts that these two known cases exhaust all possible tight codes. The paper verifies the conjecture only for n≤4,000,000, so the general classification remains unresolved.