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

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.