How this code was found
[[144,2,12]] trivariate bicycle (multivariate-bicycle family)
Primary reference: arXiv:2406.19151 (Voss, Sim, Haug, Bharti), SM Table 2, weight-4 row.
Direction & hypothesis
Among the paper's weight-4 rows, [[144,2,12]] offers the best distance at fixed check weight 4 (d = 12) and the best efficiency kd^2/n = 2·144/144 = 2.0 for low-k weight-4 designs, matching the toric [[64,2,8]] ratio (2·64/64 = 2.0) but with 50% more distance per qubit-axis. The weight-4 cell had no d >= 12 entry at any n <= 144, so the row was expected to be undominated there.
What was searched
Reproduction of the published row (l, m) = (8, 9), A = x^3 + y^7, B = x + 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.
This row uses only x/y monomials (the z variable is idle): x^3 → (3, 0), y^7 → (0, 7), x → (1, 0), y^5 → (0, 5), giving A = {(3,0), (0,7)}, B = {(1,0), (0,5)} on Z_8 × Z_9.
Evidence trail
Building H_X, H_Z from those exponent pairs yields n = 144, k = 2 (kit compute_k, matches the paper). CSS ✓. Fresh witness search (surrogate, 8000 trials per side): X logical weight 12, Z logical weight 12, both verified against the verifier's criteria. Claimed d = 12 is a witness- backed upper bound, confidence: upper_bound, matching the paper's stated d. Gate result: passed: true, "advances the weight-4 x unrestricted board".
Dead ends
The paper's other weight-4 rows were examined against the current board and either duplicate parameters of the seeded toric [[64,2,8]] (a distinct code — fingerprint differs) or lie strictly inside [[112,2,10]]/[[144,2,12]]. Weight-5/6/7 rows, while rate-friendly, do not enter the weight-4 cell.
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=8, m=9: x^3 -> (3,0), y^7 -> (0,7), x -> (1,0), y^5 -> (0,5) A_terms = [(3,0), (0,7)] B_terms = [(1,0), (0,5)] # 2. build the periodic BB matrix with the kit's own builder from bb import build_bb HX, HZ = build_bb(8, 9, 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.
Parity checks
X-checks 72 (max weight 4) · Z-checks 72 (max weight 4)
H_X (72 checks, sparse supports)
[7, 27, 77, 81]
[8, 28, 78, 82]
[0, 29, 79, 83]
[1, 30, 80, 84]
[2, 31, 72, 85]
[3, 32, 73, 86]
[4, 33, 74, 87]
[5, 34, 75, 88]
[6, 35, 76, 89]
[16, 36, 86, 90]
[17, 37, 87, 91]
[9, 38, 88, 92]
[10, 39, 89, 93]
[11, 40, 81, 94]
[12, 41, 82, 95]
[13, 42, 83, 96]
[14, 43, 84, 97]
[15, 44, 85, 98]
[25, 45, 95, 99]
[26, 46, 96, 100]
[18, 47, 97, 101]
[19, 48, 98, 102]
[20, 49, 90, 103]
[21, 50, 91, 104]
[22, 51, 92, 105]
[23, 52, 93, 106]
[24, 53, 94, 107]
[34, 54, 104, 108]
[35, 55, 105, 109]
[27, 56, 106, 110]
[28, 57, 107, 111]
[29, 58, 99, 112]
[30, 59, 100, 113]
[31, 60, 101, 114]
[32, 61, 102, 115]
[33, 62, 103, 116]
[43, 63, 113, 117]
[44, 64, 114, 118]
[36, 65, 115, 119]
[37, 66, 116, 120]
[38, 67, 108, 121]
[39, 68, 109, 122]
[40, 69, 110, 123]
[41, 70, 111, 124]
[42, 71, 112, 125]
[0, 52, 122, 126]
[1, 53, 123, 127]
[2, 45, 124, 128]
[3, 46, 125, 129]
[4, 47, 117, 130]
[5, 48, 118, 131]
[6, 49, 119, 132]
[7, 50, 120, 133]
[8, 51, 121, 134]
[9, 61, 131, 135]
[10, 62, 132, 136]
[11, 54, 133, 137]
[12, 55, 134, 138]
[13, 56, 126, 139]
[14, 57, 127, 140]
[15, 58, 128, 141]
[16, 59, 129, 142]
[17, 60, 130, 143]
[18, 70, 72, 140]
[19, 71, 73, 141]
[20, 63, 74, 142]
[21, 64, 75, 143]
[22, 65, 76, 135]
[23, 66, 77, 136]
[24, 67, 78, 137]
[25, 68, 79, 138]
[26, 69, 80, 139]
H_Z (72 checks, sparse supports)
[4, 63, 74, 117]
[5, 64, 75, 118]
[6, 65, 76, 119]
[7, 66, 77, 120]
[8, 67, 78, 121]
[0, 68, 79, 122]
[1, 69, 80, 123]
[2, 70, 72, 124]
[3, 71, 73, 125]
[0, 13, 83, 126]
[1, 14, 84, 127]
[2, 15, 85, 128]
[3, 16, 86, 129]
[4, 17, 87, 130]
[5, 9, 88, 131]
[6, 10, 89, 132]
[7, 11, 81, 133]
[8, 12, 82, 134]
[9, 22, 92, 135]
[10, 23, 93, 136]
[11, 24, 94, 137]
[12, 25, 95, 138]
[13, 26, 96, 139]
[14, 18, 97, 140]
[15, 19, 98, 141]
[16, 20, 90, 142]
[17, 21, 91, 143]
[18, 31, 72, 101]
[19, 32, 73, 102]
[20, 33, 74, 103]
[21, 34, 75, 104]
[22, 35, 76, 105]
[23, 27, 77, 106]
[24, 28, 78, 107]
[25, 29, 79, 99]
[26, 30, 80, 100]
[27, 40, 81, 110]
[28, 41, 82, 111]
[29, 42, 83, 112]
[30, 43, 84, 113]
[31, 44, 85, 114]
[32, 36, 86, 115]
[33, 37, 87, 116]
[34, 38, 88, 108]
[35, 39, 89, 109]
[36, 49, 90, 119]
[37, 50, 91, 120]
[38, 51, 92, 121]
[39, 52, 93, 122]
[40, 53, 94, 123]
[41, 45, 95, 124]
[42, 46, 96, 125]
[43, 47, 97, 117]
[44, 48, 98, 118]
[45, 58, 99, 128]
[46, 59, 100, 129]
[47, 60, 101, 130]
[48, 61, 102, 131]
[49, 62, 103, 132]
[50, 54, 104, 133]
[51, 55, 105, 134]
[52, 56, 106, 126]
[53, 57, 107, 127]
[54, 67, 108, 137]
[55, 68, 109, 138]
[56, 69, 110, 139]
[57, 70, 111, 140]
[58, 71, 112, 141]
[59, 63, 113, 142]
[60, 64, 114, 143]
[61, 65, 115, 135]
[62, 66, 116, 136]