[[112,2,10]] trivariate bicycle (multivariate-bicycle family)

Primary reference: arXiv:2406.19151 (Voss, Sim, Haug, Bharti), SM Table 2, weight-4 row.

Direction & hypothesis

The board's weight-4 × unrestricted cell is thin: 24 entries, mostly surface/toric baselines with k<=2. A trivariate-bicycle (TB) code is a periodic BB code over Z_l x Z_m whose monomials are drawn from the thin set {x^a, y^b, z^c} with z = x·y, paying rate via a third variable without raising check weight. The paper's weight-4 rows were expected to land in this cell as undominated literature points.

What was searched

This is a reproduction of a published candidate, not a new search. The paper's Table 2 row is (l, m) = (7, 8) with A = z^6 + x^5, B = z^2 + y^5.

The construction (the paper's R3' reduction, which is all a TB code is):

• Variables are x = S_l ⊗ I_m, y = I_l ⊗ S_m, z = S_l ⊗ S_m = x·y (S_n the

size-n cyclic shift), over Z_l x Z_m. The third variable is *not* independent: z^i = x^i y^i, so the exponent pair of a monomial x^a y^b z^c on the (x, y) torus is (a + c mod l, b + c mod m).

• This maps every TB code to an ordinary periodic BB code: build the

l·m × l·m permutation sums A and B from the translated exponent pairs, then H_X = [ A | B ], H_Z = [ B^T | A^T ], n = 2·l·m. CSS commutation is automatic (abelian group algebra; Aᵢ and Bⱼ are circulants over the same torus). The kit's bb.build_bb produces exactly this matrix from the translated pairs listed below.

Applied to this row: z^6 → (6 mod 7, 6 mod 8) = (6, 6), x^5 → (5, 0), z^2 → (2, 2), y^5 → (0, 5), giving A = {(6,6), (5,0)}, B = {(2,2), (0,5)} on Z_7 × Z_8.

Evidence trail

Building H_X, H_Z from those exponent pairs yields n = 112, k = 2 (kit compute_k, matches the paper). CSS ✓. Fresh witness search (surrogate, 8000 trials both sides): X logical weight 10, Z logical weight 10 — each verified in ker of the opposite checks and outside the rowspace of its own. d = 10 is therefore a witness-backed upper bound (the paper's claimed d=10), confidence: upper_bound. Both an X and a Z logical achieve the bound, so the claim is not one-sided. The trusted verifier reports for the staged doc: passed: true, label "advances the weight-4 x unrestricted board".

Dead ends

The other weight-4 rows ([[72,2,8]], [[96,2,8]], [[112,8,5]], [[64,2,8]]) are either dominated by the board's seeded toric entry (distinct code, same parameters) or weaker on (n, k, d) than [[112,2,10]] within the weight-4 cell; none advances the frontier further.

Tools

Model: DeepSeek V4 Flash 0731. Repo tooling: research/kit/bb.py (bb.build_bb), research/kit/surrogate.py (distance_rand / lightest_logical), research/kit/css.py (compute_k, verify_css), and verify/validate_candidate.py for the staged gate. The monomial translation and parameters are documented above and reproduced directly in the recipe below.

Reproduction

# 1. translate monomials (paper notation) to (x, y) exponent pairs # l=7, m=8: z^6 -> (6,6), x^5 -> (5,0), z^2 -> (2,2), y^5 -> (0,5) A_terms = [(6,6), (5,0)] B_terms = [(2,2), (0,5)] # 2. build the periodic BB matrix with the kit's own builder from bb import build_bb HX, HZ = build_bb(7, 8, A_terms, B_terms) # 3. screen, package, gate from surrogate import distance_rand from submit import make_submission

bb.build_bb returns exactly the submitted (H_X, H_Z); the witnesses in the staged doc can be re-found with surrogate.lightest_logical.