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[[288,32,8]] d ≤
n
288
k
32
d
8
kd²/n
7.111
w
6

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Distance

d_X 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[3, 11, 51, 59, 180, 188, 228, 236]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[38, 86, 101, 134, 241, 245, 249, 278]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @cbjuan
provenance submitted through the challenge
novelty novelty not audited
construction bivariate bicycle (A=[(0,0),(1,3),(5,3)], B=[(0,0),(3,1),(3,5)]) on (ell,m)=(12,12)
model Claude Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview (claimed, not verified)
date 2026-08-05
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[288,32,8]] — mixed-monomial BB, MILP-exact d=8

Direction & hypothesis

Same Campaign 4 mixed-monomial ansatz as [notes/144-16-8.md](144-16-8.md), swept to the larger (12,12) lattice to look for higher-k codes at the same distance. Found by our LLM-guided evolutionary search (arXiv:2606.02418, Campaign 4 ensemble: Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview).

What was searched

300 iterations x 750 population, mixed-monomial ansatz at (ell,m)=(12,12). This code: A(x,y) = 1 + xy^3 + x^5y^3, B(x,y) = 1 + x^3y + x^3y^5 over F_2[x,y]/(x^12-1, y^12-1) — B is A with x and y swapped in each non-constant term, a symmetric construction the ansatz converged to independently at several (ell,m) values.

Evidence trail

MILP-exact (evaluation/distance_milp.py) proved d=8 exactly: all 64 logical operators checked, 0 unproven incumbents, ~566s wall-clock. Claim: exact, d=8.

Dead ends

Non-symmetric perturbations of this A/B pair (breaking the x<->y swap symmetry between the two polynomials) at the same lattice mostly reduced d to 4-6 under MILP; higher-k relatives at (12,12) in this same family top out around k=48 at d=6 (see [notes/288-48-6.md](288-48-6.md)) — the rate-distance envelope our paper documents for indecomposable CSS codes at this block length.

Tools

Claude Opus 4.6 + GPT-5.3-Codex + Gemini 3.1 Pro Preview ensemble, openevolve harness, evaluation/bb_code.py + evaluation/distance_milp.py from qcode-discovery. 64-core server.

Reproduction

Bivariate bicycle at (ell,m)=(12,12): A = {(0,0),(1,3),(5,3)}, B = {(0,0),(3,1),(3,5)}. Construction string matches codes/288-32-8.json.

Parity checks

X-checks 144 · Z-checks 144
H_X (144 checks, sparse supports)
[0, 15, 63, 144, 181, 185] [1, 16, 64, 145, 182, 186] [2, 17, 65, 146, 183, 187] [3, 18, 66, 147, 184, 188] [4, 19, 67, 148, 185, 189] [5, 20, 68, 149, 186, 190] [6, 21, 69, 150, 187, 191] [7, 22, 70, 151, 180, 188] [8, 23, 71, 152, 181, 189] [9, 12, 60, 153, 182, 190] [10, 13, 61, 154, 183, 191] [11, 14, 62, 155, 180, 184] [12, 27, 75, 156, 193, 197] [13, 28, 76, 157, 194, 198] [14, 29, 77, 158, 195, 199] [15, 30, 78, 159, 196, 200] [16, 31, 79, 160, 197, 201] [17, 32, 80, 161, 198, 202] [18, 33, 81, 162, 199, 203] [19, 34, 82, 163, 192, 200] [20, 35, 83, 164, 193, 201] [21, 24, 72, 165, 194, 202] [22, 25, 73, 166, 195, 203] [23, 26, 74, 167, 192, 196] [24, 39, 87, 168, 205, 209] [25, 40, 88, 169, 206, 210] [26, 41, 89, 170, 207, 211] [27, 42, 90, 171, 208, 212] [28, 43, 91, 172, 209, 213] [29, 44, 92, 173, 210, 214] [30, 45, 93, 174, 211, 215] [31, 46, 94, 175, 204, 212] [32, 47, 95, 176, 205, 213] [33, 36, 84, 177, 206, 214] [34, 37, 85, 178, 207, 215] [35, 38, 86, 179, 204, 208] [36, 51, 99, 180, 217, 221] [37, 52, 100, 181, 218, 222] [38, 53, 101, 182, 219, 223] [39, 54, 102, 183, 220, 224] [40, 55, 103, 184, 221, 225] [41, 56, 104, 185, 222, 226] [42, 57, 105, 186, 223, 227] [43, 58, 106, 187, 216, 224] [44, 59, 107, 188, 217, 225] [45, 48, 96, 189, 218, 226] [46, 49, 97, 190, 219, 227] [47, 50, 98, 191, 216, 220] [48, 63, 111, 192, 229, 233] [49, 64, 112, 193, 230, 234] [50, 65, 113, 194, 231, 235] [51, 66, 114, 195, 232, 236] [52, 67, 115, 196, 233, 237] [53, 68, 116, 197, 234, 238] [54, 69, 117, 198, 235, 239] [55, 70, 118, 199, 228, 236] [56, 71, 119, 200, 229, 237] [57, 60, 108, 201, 230, 238] [58, 61, 109, 202, 231, 239] [59, 62, 110, 203, 228, 232] [60, 75, 123, 204, 241, 245] [61, 76, 124, 205, 242, 246] [62, 77, 125, 206, 243, 247] [63, 78, 126, 207, 244, 248] [64, 79, 127, 208, 245, 249] [65, 80, 128, 209, 246, 250] [66, 81, 129, 210, 247, 251] [67, 82, 130, 211, 240, 248] [68, 83, 131, 212, 241, 249] [69, 72, 120, 213, 242, 250] [70, 73, 121, 214, 243, 251] [71, 74, 122, 215, 240, 244] [72, 87, 135, 216, 253, 257] [73, 88, 136, 217, 254, 258] [74, 89, 137, 218, 255, 259] [75, 90, 138, 219, 256, 260] [76, 91, 139, 220, 257, 261] [77, 92, 140, 221, 258, 262] [78, 93, 141, 222, 259, 263] [79, 94, 142, 223, 252, 260] [80, 95, 143, 224, 253, 261] [81, 84, 132, 225, 254, 262] [82, 85, 133, 226, 255, 263] [83, 86, 134, 227, 252, 256] [3, 84, 99, 228, 265, 269] [4, 85, 100, 229, 266, 270] [5, 86, 101, 230, 267, 271] [6, 87, 102, 231, 268, 272] [7, 88, 103, 232, 269, 273] [8, 89, 104, 233, 270, 274] [9, 90, 105, 234, 271, 275] [10, 91, 106, 235, 264, 272] [11, 92, 107, 236, 265, 273] [0, 93, 96, 237, 266, 274] [1, 94, 97, 238, 267, 275] [2, 95, 98, 239, 264, 268] [15, 96, 111, 240, 277, 281] [16, 97, 112, 241, 278, 282] [17, 98, 113, 242, 279, 283] [18, 99, 114, 243, 280, 284] [19, 100, 115, 244, 281, 285] [20, 101, 116, 245, 282, 286] [21, 102, 117, 246, 283, 287] [22, 103, 118, 247, 276, 284] [23, 104, 119, 248, 277, 285] [12, 105, 108, 249, 278, 286] [13, 106, 109, 250, 279, 287] [14, 107, 110, 251, 276, 280] [27, 108, 123, 145, 149, 252] [28, 109, 124, 146, 150, 253] [29, 110, 125, 147, 151, 254] [30, 111, 126, 148, 152, 255] [31, 112, 127, 149, 153, 256] [32, 113, 128, 150, 154, 257] [33, 114, 129, 151, 155, 258] [34, 115, 130, 144, 152, 259] [35, 116, 131, 145, 153, 260] [24, 117, 120, 146, 154, 261] [25, 118, 121, 147, 155, 262] [26, 119, 122, 144, 148, 263] [39, 120, 135, 157, 161, 264] [40, 121, 136, 158, 162, 265] [41, 122, 137, 159, 163, 266] [42, 123, 138, 160, 164, 267] [43, 124, 139, 161, 165, 268] [44, 125, 140, 162, 166, 269] [45, 126, 141, 163, 167, 270] [46, 127, 142, 156, 164, 271] [47, 128, 143, 157, 165, 272] [36, 129, 132, 158, 166, 273] [37, 130, 133, 159, 167, 274] [38, 131, 134, 156, 160, 275] [3, 51, 132, 169, 173, 276] [4, 52, 133, 170, 174, 277] [5, 53, 134, 171, 175, 278] [6, 54, 135, 172, 176, 279] [7, 55, 136, 173, 177, 280] [8, 56, 137, 174, 178, 281] [9, 57, 138, 175, 179, 282] [10, 58, 139, 168, 176, 283] [11, 59, 140, 169, 177, 284] [0, 48, 141, 170, 178, 285] [1, 49, 142, 171, 179, 286] [2, 50, 143, 168, 172, 287]
H_Z (144 checks, sparse supports)
[0, 115, 119, 144, 237, 285] [1, 108, 116, 145, 238, 286] [2, 109, 117, 146, 239, 287] [3, 110, 118, 147, 228, 276] [4, 111, 119, 148, 229, 277] [5, 108, 112, 149, 230, 278] [6, 109, 113, 150, 231, 279] [7, 110, 114, 151, 232, 280] [8, 111, 115, 152, 233, 281] [9, 112, 116, 153, 234, 282] [10, 113, 117, 154, 235, 283] [11, 114, 118, 155, 236, 284] [12, 127, 131, 153, 156, 249] [13, 120, 128, 154, 157, 250] [14, 121, 129, 155, 158, 251] [15, 122, 130, 144, 159, 240] [16, 123, 131, 145, 160, 241] [17, 120, 124, 146, 161, 242] [18, 121, 125, 147, 162, 243] [19, 122, 126, 148, 163, 244] [20, 123, 127, 149, 164, 245] [21, 124, 128, 150, 165, 246] [22, 125, 129, 151, 166, 247] [23, 126, 130, 152, 167, 248] [24, 139, 143, 165, 168, 261] [25, 132, 140, 166, 169, 262] [26, 133, 141, 167, 170, 263] [27, 134, 142, 156, 171, 252] [28, 135, 143, 157, 172, 253] [29, 132, 136, 158, 173, 254] [30, 133, 137, 159, 174, 255] [31, 134, 138, 160, 175, 256] [32, 135, 139, 161, 176, 257] [33, 136, 140, 162, 177, 258] [34, 137, 141, 163, 178, 259] [35, 138, 142, 164, 179, 260] [7, 11, 36, 177, 180, 273] [0, 8, 37, 178, 181, 274] [1, 9, 38, 179, 182, 275] [2, 10, 39, 168, 183, 264] [3, 11, 40, 169, 184, 265] [0, 4, 41, 170, 185, 266] [1, 5, 42, 171, 186, 267] [2, 6, 43, 172, 187, 268] [3, 7, 44, 173, 188, 269] [4, 8, 45, 174, 189, 270] [5, 9, 46, 175, 190, 271] [6, 10, 47, 176, 191, 272] [19, 23, 48, 189, 192, 285] [12, 20, 49, 190, 193, 286] [13, 21, 50, 191, 194, 287] [14, 22, 51, 180, 195, 276] [15, 23, 52, 181, 196, 277] [12, 16, 53, 182, 197, 278] [13, 17, 54, 183, 198, 279] [14, 18, 55, 184, 199, 280] [15, 19, 56, 185, 200, 281] [16, 20, 57, 186, 201, 282] [17, 21, 58, 187, 202, 283] [18, 22, 59, 188, 203, 284] [31, 35, 60, 153, 201, 204] [24, 32, 61, 154, 202, 205] [25, 33, 62, 155, 203, 206] [26, 34, 63, 144, 192, 207] [27, 35, 64, 145, 193, 208] [24, 28, 65, 146, 194, 209] [25, 29, 66, 147, 195, 210] [26, 30, 67, 148, 196, 211] [27, 31, 68, 149, 197, 212] [28, 32, 69, 150, 198, 213] [29, 33, 70, 151, 199, 214] [30, 34, 71, 152, 200, 215] [43, 47, 72, 165, 213, 216] [36, 44, 73, 166, 214, 217] [37, 45, 74, 167, 215, 218] [38, 46, 75, 156, 204, 219] [39, 47, 76, 157, 205, 220] [36, 40, 77, 158, 206, 221] [37, 41, 78, 159, 207, 222] [38, 42, 79, 160, 208, 223] [39, 43, 80, 161, 209, 224] [40, 44, 81, 162, 210, 225] [41, 45, 82, 163, 211, 226] [42, 46, 83, 164, 212, 227] [55, 59, 84, 177, 225, 228] [48, 56, 85, 178, 226, 229] [49, 57, 86, 179, 227, 230] [50, 58, 87, 168, 216, 231] [51, 59, 88, 169, 217, 232] [48, 52, 89, 170, 218, 233] [49, 53, 90, 171, 219, 234] [50, 54, 91, 172, 220, 235] [51, 55, 92, 173, 221, 236] [52, 56, 93, 174, 222, 237] [53, 57, 94, 175, 223, 238] [54, 58, 95, 176, 224, 239] [67, 71, 96, 189, 237, 240] [60, 68, 97, 190, 238, 241] [61, 69, 98, 191, 239, 242] [62, 70, 99, 180, 228, 243] [63, 71, 100, 181, 229, 244] [60, 64, 101, 182, 230, 245] [61, 65, 102, 183, 231, 246] [62, 66, 103, 184, 232, 247] [63, 67, 104, 185, 233, 248] [64, 68, 105, 186, 234, 249] [65, 69, 106, 187, 235, 250] [66, 70, 107, 188, 236, 251] [79, 83, 108, 201, 249, 252] [72, 80, 109, 202, 250, 253] [73, 81, 110, 203, 251, 254] [74, 82, 111, 192, 240, 255] [75, 83, 112, 193, 241, 256] [72, 76, 113, 194, 242, 257] [73, 77, 114, 195, 243, 258] [74, 78, 115, 196, 244, 259] [75, 79, 116, 197, 245, 260] [76, 80, 117, 198, 246, 261] [77, 81, 118, 199, 247, 262] [78, 82, 119, 200, 248, 263] [91, 95, 120, 213, 261, 264] [84, 92, 121, 214, 262, 265] [85, 93, 122, 215, 263, 266] [86, 94, 123, 204, 252, 267] [87, 95, 124, 205, 253, 268] [84, 88, 125, 206, 254, 269] [85, 89, 126, 207, 255, 270] [86, 90, 127, 208, 256, 271] [87, 91, 128, 209, 257, 272] [88, 92, 129, 210, 258, 273] [89, 93, 130, 211, 259, 274] [90, 94, 131, 212, 260, 275] [103, 107, 132, 225, 273, 276] [96, 104, 133, 226, 274, 277] [97, 105, 134, 227, 275, 278] [98, 106, 135, 216, 264, 279] [99, 107, 136, 217, 265, 280] [96, 100, 137, 218, 266, 281] [97, 101, 138, 219, 267, 282] [98, 102, 139, 220, 268, 283] [99, 103, 140, 221, 269, 284] [100, 104, 141, 222, 270, 285] [101, 105, 142, 223, 271, 286] [102, 106, 143, 224, 272, 287]
raw submission JSON