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[[288,48,6]] d ≤
n
288
k
48
d
6
kd²/n
6.0
w
6

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Distance

d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[3, 29, 49, 55, 81, 101]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[7, 29, 51, 57, 73, 101]
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),(2,4),(4,2)], B=[(0,0),(8,2),(10,4)]) 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

[[288,48,6]] — mixed-monomial BB, MILP-exact d=6, high rate (k/n=1/6)

Direction & hypothesis

Campaign 4 mixed-monomial sweep at (12,12) targeting the high-rate end of the family in [notes/288-32-8.md](288-32-8.md), trading distance for k. 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 + x^2y^4 + x^4y^2, B(x,y) = 1 + x^8y^2 + x^10y^4 over F_2[x,y]/(x^12-1, y^12-1) — same x<->y-swap-symmetric structure as [[288,32,8]] but with doubled exponents, trading 2 units of distance for 16 more logical qubits.

Evidence trail

Independent BP-OSD_0 batches on this code returned d>=8 across multiple 5,000-trial runs (one batch reported d_symplectic=8); as with other codes in this batch, a decoder upper bound alone doesn't settle the exact value, so we turned to MILP. MILP-exact (evaluation/distance_milp.py) proved d=6 exactly: all 96 logical operators checked, 0 unproven incumbents, ~149s wall-clock. Claim: exact, d=6.

Dead ends

Pushing the exponent scaling further (beyond the x2 relative to [[288,32,8]] used here) collapsed distance to 4 or 2 at the same (ell,m); this sits near the edge of the rate-distance envelope our paper reports for indecomposable codes at n=288 (k>48 implies d<=4 in our full catalog 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),(2,4),(4,2)}, B = {(0,0),(8,2),(10,4)}. Construction string matches codes/288-48-6.json.

Parity checks

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