Power-of-two lexicode construction and Catalan clique covers

Establish that, for every power of two k greater than 1, the lexicode in the Johnson graph J(2k, k−1) has cardinality one-half of the kth Catalan number C_k and consists of pairwise-overlapping elements; consequently, construct an independent set of cardinality C_k in JK(2k, k−1) and prove that the clique covering number satisfies θ(J(2k, k)) = C_k.

Background

For even parameters of the form N = 2k, the paper relates clique covers of the Johnson graph J(2k, k) to independent sets in the auxiliary graph JK(2k, k−1). A lexicographic constant-weight binary code, or lexicode, in J(2k, k−1) can be duplicated across the two parts of JK(2k, k−1) when its codewords are pairwise overlapping. If the lexicode has one-half of the Catalan number C_k of codewords, this duplication yields an independent set of size C_k and implies θ(J(2k, k)) = C_k.

The construction is verified computationally for k = 2, 4, 8, and 16, and the paper proves the corresponding clique-cover values for k = 8 and k = 16. The authors propose extending this observed power-of-two pattern to every power of two greater than 1.

References

Given this evidence above, we conjecture that this construction works whenever $k$ is a power of $2$. Assume that $k > 1$ is a power of $2$. The lexicode in $J(2k, k-1)$ has cardinality $\tfrac{1}{2} C_k$, and its elements are pairwise overlapping. In particular, the code defines an independent set of cardinality $C_k$ in $JK(2k, k-1)$ so that $\theta(J(2k, k)) = C_k$.

— On the clique covering numbers of Johnson graphs  (2502.15019 - Jørgensen, 20 Feb 2025) in Remark immediately preceding Conjecture 4.?, Section 4, “Bounds from coding theory”; Conjecture following the remark