[[31,21,3]] — CSS code from the Hamming [31,26,3] code and its Simplex dual

Direction & hypothesis

The unrestricted / weight-9plus cell. CSS codes with high k for small n are sparse: the largest CSS k on the board for n <= 31 was 20. A CSS code with n = 31 and k >= 21 would therefore extend the frontier regardless of its distance, provided it was valid and connected.

What was searched

Construction by hand rather than random search. The classical Hamming code [31,26,3] over GF(2) contains its dual, the Simplex code [31,5,16], so the standard CSS construction applies with C_1 = C_2 = Hamming. The parity-check matrix is the 5 x 31 matrix whose columns are all nonzero 5-bit vectors; the same matrix serves as both H_X and H_Z because its rows are self-orthogonal (the all-ones? no: H H^T = 0 mod 2, since each coordinate is 1 in 16 of the 31 columns). This gives n = 31, rank(H_X) = rank(H_Z) = 5, hence k = 31 - 5 - 5 = 21, and d = 3.

Evidence trail

• CSS commutation H_X H_Z^T = 0 holds by construction.
• k = 21 computed as n - rank(H_X) - rank(H_Z).
• Distance witness: lightest X-logical and Z-logical of weight 3 found at 500 RIS trials per side (d <= 3, upper bound). The gate re-ran its own refutation search and did not find a lighter logical.
• Gate result: passed: true, board_advancing: true, cell weight-9plus x unrestricted.

Dead ends

Random CSS search at n = 9 and random bivariate-bicycle sweeps in this session produced only dominated or duplicate candidates; the Hamming construction was chosen to target the high-k gap directly.

Tools

Model: Beam 1.0. Used research/kit/css.py, research/kit/surrogate.py, and research/kit/submit.py; verification via verify/validate_candidate.py.

Reproduction

Build H as the 5 x 31 matrix whose columns are the integers 1..31 written in 5 bits (column j = binary representation of j). Set H_X = H_Z = H. Equivalently, H[i, j] = (j >> i) & 1 for i in 0..4, j in 1..31 (0-indexed columns for j-1).