New lower bounds for entries in A. E. Brouwer’s tables of binary, binary constant-weight, ternary, mixed binary/ternary and 5-ary codes, a proposed table of spectral-signature codes, and the search scripts. Oscar Sanchez (GitHub: owhs), October 2026, with Claude (Anthropic).
Each result folder has the codes as plain text and a checker that uses only the Python standard library.
| folder | result | table value |
|---|---|---|
binary_A18_4/ |
A(18,4) = A(17,3) >= 6144 and A(17,4) = A(16,3) >= 3072 (binary table) | 5632, 2816 |
constant_weight_A31_10_12_1015/ |
A(31,10,12) >= 1015 | 961 |
constant_weight_A31_10_12/ |
A(31,10,12) >= 961 (earlier code, superseded by 1015; on Brouwer’s page since 2026-10-14) | 930 before |
constant_weight_A32_10_10/ |
A(32,10,10) >= 558 | 500 |
ternary_mixed/ |
27 codes (25 found by search, 2 derived from them), for example A_3(15,5) >= 7929, A_3(14,8) >= 123, N(5,8,5) >= 513, N(10,4,4) >= 1104 | 7812, 108, 486, 1050 |
quinary/A5_10_4_18750/ |
A_5(10,4) >= 18750 | 17500 |
quinary/A5_9_4_4500/ |
A_5(9,4) >= 4500 | 4375 |
quinary/A5_7_3_1601/ |
A_5(7,3) >= 1601 | 1597 |
quinary/A5_9_5_893/ |
A_5(9,5) >= 893, by repacking Laaksonen and Ostergard’s 857 code (their 2019 value); checkers need numpy | 725 |
quinary/reproductions_LO2019/ |
independent codes for A_5(8,5) >= 257 and A_5(7,5) >= 57, and A_5(7,4) >= 257 by deleting a coordinate of the first; these values are due to Laaksonen and Ostergard (2019, Table 3) and are not yet on the 5-ary page | 165, 53, 250 |
derived_from_LO2019/ |
five mixed binary/ternary bounds that follow from Laaksonen and Ostergard’s ternary codes, for example N(1,13,4) >= 18711; not yet on the mixed page | 16767 |
spectral_signature/ |
proposed table: exact sizes of binary constant-weight codes whose words have distinct power spectra, n <= 20 | none |
search_code/ |
the search and checking scripts |
The table values were read from Brouwer’s pages on 2026-10-16. The mixed codes were compared with Brouwer’s page only; the 5-ary values were also compared with Table 3 of Laaksonen and Ostergard (2019) and with Laaksonen’s results page.
Related: DNA_GC_codes_2026, and the Game of Sloanes pull request https://github.com/gnikylime/GameofSloanes/pull/18.
A group of coordinate symmetries is fixed, the orbits of words that are valid codes are enumerated, and orbits are chosen by a clique search
(prescribed-automorphism search). Details and references: METHOD_prescribed_automorphism_search.md.