Improving the covering code bound K(1,7,1) ≤ 48 requires an asymmetric code
Year: 2026
Type: preprint
Abstract: Let K(1, 7, 1) be the smallest number of words in a covering code of radius 1 in the mixed Hamming space F3×F72. The tables of mixed binary/ternary covering codes give 42 ≤ K(1, 7, 1) ≤ 48, and the upper bound comes from the direct product of F3 with the binary Hamming code of length 7. We prove, with the help of a computer, that every covering code of radius 1 in this space with at most 47 words has trivial automorphism group. The proof reduces the question to one representative of each of the 46 conjugacy classes of isometries of prime order and solves an integer program for each; the minimum size of an invariant covering code turns out to be 48, 54 or 56 in every case. Two independent integer-programming solvers agree on all 46 cases.
Author: Parsons, Noah
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225594947