Catalan Bounds for Symmetric Strength‐Two Orthogonal Arrays

ABSTRACT A Hamming shell construction is a two‐level array obtained by taking every binary vector of a given Hamming weight a prescribed number of times, for each weight in turn. Such arrays are invariant under all permutations of the factors, and they are strength‐two orthogonal arrays exactly when the multiplicities satisfy three linear constraints. This article determines how small such an array can be. The main result is a sharp upper bound on the smallest run size, equal to four times a central binomial coefficient, attained for every number of factors by an explicit construction whose multiplicities are Catalan numbers; for an even number of factors, the extremal array is the complete design of all half‐size subsets, supplemented by Catalan many copies of the empty set and of the full set. The bound is conjectured to be exact, and this is verified by exhaustive enumeration for up to 14 factors. Two structural results accompany the bound: a closed‐form expression for the K‐aberration functional of Mukerjee and Tang as a positive‐definite quadratic form in adjacent entries of the count vector, and a self‐complementary (level‐flip) symmetry that leaves the second‐order entry of the functional invariant while breaking the symmetry of all higher orders. Consequences for fractional factorial designs under the baseline parameterisation are discussed.