An explicit (54,13,4)-covering with 665 blocks

We give an explicit family of 665 distinct 13-subsets of a 54-element set containing every 4-subset. This establishes C(54,13,4) <= 665, improving the upper bound 666 reported by Van Staden and listed in the Covering Repository as checked on 6 November 2026. The lower bound 536 is unchanged; no exact value is claimed. The construction uses Wynand van Staden’s existing weighted local-search implementation, starting from an archived 746-block design credited to Jan de Heer and Steve Muir. The contribution is a one-block numerical improvement, not a new algorithm. Novelty is qualified by the public-source search; unindexed or unpublished earlier results cannot be excluded. Files include the four-page preprint, complete block list, LaTeX source, and a supplementary archive with exact exhaustive checkers, the original MIT-licensed search code, a deterministic replay wrapper, input data, logs, attribution, and checksums. All 316251 quadruples are covered. Certificate SHA-256: b924cd72711ec77a9c1cca6aba658fc14d3fa5a0f684d28ccc8aeb5260768162. OpenAI Codex was used substantially for literature searching, experiment orchestration, verification code and execution, and drafting. AI assistance is disclosed in the paper; AI is not listed as an author. This is a preprint, not a peer-reviewed publication. The paper, new certificate, and documentation are licensed CC BY 4.0. Software retains the MIT licenses supplied in the archive. The starting design retains the attribution to Gordon’s La Jolla Covering Repository archive (https://zenodo.org/records/19735294). Prior work: https://doi.org/10.20944/preprints202610.2905.v1; search implementation: https://github.com/WynandVStaden/covering-design-records.