🤖 AI Summary
This work addresses the lack of formal verification for upper bounds, lower bounds, and exact covering numbers in q-ary covering codes by systematically formalizing their foundational theory in Lean 4. The formalization includes the Hamming ball volume formula, sphere-covering lower bounds, product rules, and exact covering certificates for small instances. We introduce provable certificate predicates to establish an end-to-end pipeline from classical literature results to machine-verifiable proofs, and construct the first proof-carrying database for covering codes that supports automatic replay validation. Our implementation successfully reproduces and verifies selected upper bounds originally established by van Laarhoven et al. (1989), with all bounds accompanied by traceable, auditable Lean proof artifacts, thereby providing a reusable and formally verified foundation for coding theory.
📝 Abstract
Covering codes in finite Hamming spaces ask for small sets of words whose Hamming balls cover the whole space. This paper presents a Lean 4 formalization of the elementary theory of q-ary covering codes, centered on certificate predicates for upper bounds, lower bounds, and exact covering numbers $K_q(n,r)$. The formalization proves the q-ary Hamming-ball volume formula, the sphere-covering lower bound, elementary exact cases, product and relation rules, and selected small exact certificates. It also demonstrates an end-to-end workflow for checking explicit upper bounds transcribed from van Laarhoven et al. (1989). The accompanying database is proof-carrying: stored bounds have traces that replay to Lean proofs of the corresponding upper- or lower-bound predicates. The contribution is not new record bounds or a reproduction of known tables, but a reusable, auditable foundation for machine-checked covering-code certificates.