[[102,20,10]] — weight-9 generalized bicycle code over Z_51

Direction & hypothesis

Two-block codes at check weight 9 with unbalanced blocks (5 + 4), just beyond the group orders covered by the published exhaustive two-block enumeration (Lin and Pryadko, arXiv:2306.16400, abelian groups of order at most 50). The aim was a high rate at n ≈ 100 without losing distance.

What was searched

A randomized search over weight-9 two-block group-algebra codes with block weights 5 + 4 and 6 + 3, at n ≈ 100 (groups of order 45–55), with pair classes taken up to the standard equivalences. Candidates were screened for k > 0 and for low-weight logicals before any exact distance computation. Full details will appear in a forthcoming paper.

Evidence trail

• The rank gives k = 20. An independent polynomial check agrees:

x^51 - 1 factors over F_2 into factors of degrees 1, 2 and six of degree 8, and a and b share the degree-2 factor and one degree-8 factor, so k = 2 * (2 + 8) = 20.

• d_Z = 10 exact, by two independent methods:
• a connected-cluster minimum-weight search (every weight below 10

refuted, witness checked);

• dist-m4ri, exact mode.
• d_X = d_Z by the X/Z-exchange symmetry of the code.
• The matrices were rebuilt independently in GAP, and a QDistRnd run of

20,000 iterations found no logical lighter than 10.

• The claim here is witness-backed (upper bound 10 on each side); our

computations show it is exact.

Dead ends

At n ≈ 100 and weight 9, most high-k candidates collapsed to d ≤ 8. The nearest miss we found was [[98,12,12]] (kd²/n ≈ 17.6).

Tools

Our own search pipeline (Python with numba), exact distance by a connected-cluster search and by dist-m4ri, and GAP for an independent rebuild. No language model was used in the search loop.

Reproduction

Generalized bicycle code over Z_51, with a = 1 + x^19 + x^27 + x^35 + x^48 and b = 1 + x^7 + x^9 + x^19 (mod x^51 - 1). A and B are the circulants with row g supported on g + s for s in the exponents of a (resp. b); H_X = [A | B] and H_Z = [B^T | A^T].