New upper bounds on covering codes K_q(n,R) for alphabets of size six and seven

Mark Marosi

arXiv:2608.19872v1 Primary: math.CO Secondary: cs.IT

Abstract

We present improved upper bounds for nine entries of the standard tables of bounds on K_q(n,R), the minimum cardinality of a q-ary code of length n with covering radius R, for q in {6,7}: K_6(7,3)<=232, K_6(8,3)<=1045, K_6(8,4)<=167, K_6(9,4)<=703, K_6(9,5)<=123, K_6(10,4)<=2951, K_6(10,5)<=610, K_7(8,4)<=329, and K_7(9,4)<=1743. The previous best bounds, recorded in Keri's tables (last updated 2011), all arose from general constructions (direct sums and related product rules) rather than from explicit search; to our knowledge these are the first improvements to any upper bound on K_q(n,R) with q>=5 since 2011. The new bounds were found by focused local search seeded with the construction-based incumbents. All nine codes are given explicitly in the ancillary files, together with a standalone verifier; each code was checked by four independent exhaustive verification methods.

Disclosure

“(BME), Budapest, Hungary. email: marosi@mit.bme.hu. The search software, the verification protocol, the constructions, and this manuscript were produced by the AI system Claude (Anthropic), operating autonomously on a single NVIDIA GH200 node; the author’s contributions were the initiating idea, methodological guidance, a”
PDF page 1

Classification: Substantial mathematical content or result generation
Multiplier: 10
Verified: 2026-08-22T03:10:20.933413Z

Structural counts

Pages 6 pdf

Theorems 0 source

Lemmas 1 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 2 source

Bibliography entries 8 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file main.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.