A classification of invariant sets of constant-weight tuples and applications to covering schemes

Covering schemes were recently introduced as a generalization of both difference matrices and difference schemes, providing an efficient method for building covering arrays with a certain degree of symmetry. On the other hand, Tang and Woo (1983) showed that sets of constant-weight tuples yield several upper bounds on covering arrays. However, these sets usually do not have enough symmetry to induce good covering schemes. In this work, we derive a classification of invariant sets of constant-weight tuples under the action of the diagonal group of $$\mathbb {Z}_q^k$$ . This classification enables us to obtain new upper bounds on covering scheme numbers, including the exact class for binary covering scheme numbers with k columns and strength $$k-2.$$