An arXiv preprint, version 1 dated 20 August 2026, reports improved upper bounds for nine q-ary covering-code instances with alphabet sizes six and seven, after a focused computer search. The highlighted change is for K_6(8,4): the reported upper bound moves from 216 to 167, a reduction of 23%.
Put simply, the search is trying to leave no word outside a stated covering radius; a word left outside it is counted as uncovered. The notation K_q(n,R) records the alphabet setting q, the code length n and the covering radius R. The reported figures are upper bounds, not proof that the exact minimum has been found. The nine entries comprise seven q=6 instances and two q=7 instances.
The bounds behind the update
For q=6, the paper reports K_6(7,3) at most 232; K_6(8,3) at most 1,045; K_6(8,4) at most 167; K_6(9,4) at most 703; K_6(9,5) at most 123; K_6(10,4) at most 2,951; and K_6(10,5) at most 610.
The two q=7 results are K_7(8,4) at most 329 and K_7(9,4) at most 1,743. Taken together, these are nine reported upper bounds, not nine claims of optimality: a smaller code could still exist for any listed case.
The earlier comparison points were inherited from general constructions rather than produced through explicit search. The authors state that, to their knowledge, the work is the first improvement to any upper bound on K_q(n,R) for q at least 5 since 2011. That is an author-reported context claim; the direct evidence here remains limited to the nine listed instances.
A search seeded by earlier constructions
The search did not begin from a blank slate. For each instance, the initial code was rebuilt at the size of the incumbent bound, using the construction that had produced that bound. The program then selected a random uncovered word and generated candidate moves by nudging codewords toward it.
Each candidate move was evaluated exactly and in parallel. The search committed a best move, with random tie-breaking when several candidates were equally good. This sequence was the core focused-local-search step described by the paper.
After that phase, a remove-and-repair descent removed the codeword with the smallest private coverage. It used local search to repair the resulting deficit and repeated the process after a successful repair. The method combined focused local search with this repair-based descent.
All record searches ran on the 64-core Grace CPU of a single NVIDIA GH200 node.
The acknowledgments describe the pipeline's provenance. Claude autonomously designed, implemented and operated the discovery and verification pipeline, while the human author contributed the initiating idea, methodological guidance and computational resources.
The result was checked several ways
Verification used several separate checks. The paper says all nine codes were checked by four independent exhaustive verification methods. A fifth independently written dilation verifier re-confirmed all nine codes.
These checks address whether each reported code satisfies the claimed covering-radius requirement. They do not establish that its reported number is the exact minimum K_q(n,R). Verification confirms the code property; the upper-bound result leaves optimality open.
A parameter study records different results under different search settings. For K_6(6,3), exact evaluation of all candidate moves ended with zero words left uncovered. When candidates were sampled instead, the reported final uncovered counts were 82 and 18 under settings that sampled 24 or 64 candidate moves. No uncertainty estimate was reported for this comparison.
Why the numbers remain provisional
None of the searches had converged when they were stopped. The authors expect several of the bounds may improve with more computation. The nine values should therefore be read as verified upper bounds, not as final exact minima.
The evidence is also limited to nine selected instances: seven with q=6 and two with q=7. The searches had not converged when stopped, so the paper does not establish that these runs reached their best possible codes.
The materials are available for independent inspection. The nine codes, verifier and README are ancillary files of the arXiv submission, and the paper states that the full search and verification source code is public in the stated repository. The repository identified in the paper is Mapika/coldcase, under the covering-code track in cov/.
Taken together, the preprint offers a targeted update to a small set of covering-code bounds: nine new reported upper bounds, independent checks, and a parameter study showing different final uncovered counts under different search settings. The next questions are how much further computation can improve the nine bounds and how reliably the method improves other construction-inherited entries.
Paper data and sources
Original title: New upper bounds on covering codes K_q(n,R) for alphabets of size six and seven
Authors: Mark Marosi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text