[[196,2,14]] · QEC Challenge

[[196,2,14]] d ≤

196

kd²/n

2.0

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 4, w_Z = 4 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 14 · witness weight 14 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-11

witness operator (support, 14 qubits)

[43, 46, 48, 56, 57, 58, 60, 77, 138, 139, 142, 144, 156, 159]

d_Z 14 · witness weight 14 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-11-11

witness operator (support, 14 qubits)

[84, 85, 92, 98, 100, 106, 164, 165, 167, 169, 173, 180, 186, 188]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4 · H_Z 4

qubit degrees H_X 2 · H_Z 2

trapping sets H_X (1,2)×196 (2,2)×588 (3,2)×1764 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 196 (2,2): 588 (3,2): 1764 (3,4): 392

trapping sets H_Z (1,2)×196 (2,2)×588 (3,2)×1764 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 196 (2,2): 588 (3,2): 1764 (3,4): 392

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction quantum Tanner code

model Beam 1.0 (claimed, not verified)

date 2026-11-11

family quantum Tanner (a tag, not a ranking)

locality unrestricted (computed from the layout)

weight class weight ≤ 4 (computed)

How this code was found

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

[[196,2,14]] — quantum Tanner code over PSL(2,8)

Direction & hypothesis

Aimed at the weight-4 unrestricted CSS cell with quantum Tanner codes over PSL(2,8). Seed 3 gives n=196, d=14 at weight 4.

What was searched

Random quantum Tanner codes over PSL(2,8) with RepetitionCode(2), seeds 0..7, screened at 300 trials per side and re-screened at 5,000 trials per side.

Evidence trail

Witness-backed upper bound d<=14 on both X and Z sides. Full gate refutation passed on a fresh seed.

Dead ends

Several seeds gave d=14 with n=196/252; bipartite variants were weaker.

Tools

qLDPC library (QTCode.random), challenge research kit, gf2_fast. Model: Beam 1.0.

Reproduction

Group PSL(2,8), seed code repetition length 2, random seed 3, bipartite False.

Parity checks

X-checks 98 (max weight 4) · Z-checks 98 (max weight 4)

H_X (98 checks, sparse supports)

[0, 1, 2, 3] [174, 175, 176, 177] [110, 111, 112, 113] [54, 55, 56, 57] [138, 139, 140, 141] [82, 83, 84, 85] [26, 27, 28, 29] [185, 186, 187, 188] [157, 158, 159, 160] [101, 102, 103, 104] [45, 46, 47, 48] [129, 130, 131, 132] [73, 74, 75, 76] [17, 18, 19, 20] [65, 66, 67, 68] [121, 122, 123, 124] [149, 150, 151, 152] [164, 166, 167, 168] [37, 38, 39, 40] [93, 94, 95, 96] [6, 7, 8, 9] [5, 10, 11, 12] [120, 125, 126, 127] [92, 97, 98, 99] [148, 153, 154, 155] [36, 41, 42, 43] [64, 69, 70, 71] [173, 178, 179, 180] [52, 58, 59, 60] [172, 181, 182, 183] [136, 142, 143, 144] [4, 13, 14, 15] [80, 86, 87, 88] [108, 114, 115, 116] [24, 30, 31, 32] [192, 193, 194, 195] [53, 61, 62, 63] [25, 33, 34, 35] [137, 145, 146, 147] [81, 89, 90, 91] [109, 117, 118, 119] [165, 169, 170, 171] [156, 161, 162, 163] [16, 21, 22, 23] [44, 49, 50, 51] [72, 77, 78, 79] [128, 133, 134, 135] [184, 189, 190, 191] [100, 105, 106, 107] [167, 172, 173, 174] [29, 30, 36, 37] [85, 86, 92, 93] [3, 4, 5, 6] [57, 58, 64, 65] [113, 114, 120, 121] [141, 142, 148, 149] [179, 183, 184, 185] [32, 42, 44, 45] [60, 70, 72, 73] [88, 98, 100, 101] [144, 154, 156, 157] [11, 15, 16, 17] [116, 126, 128, 129] [67, 80, 81, 82] [8, 24, 25, 26] [39, 52, 53, 54] [123, 136, 137, 138] [95, 108, 109, 110] [2, 13, 164, 165] [151, 177, 181, 192] [83, 90, 187, 191] [55, 62, 159, 163] [0, 103, 107, 170] [47, 51, 139, 146] [27, 34, 131, 135] [75, 79, 175, 194] [19, 23, 111, 118] [76, 166, 171, 176] [84, 94, 119, 188] [56, 66, 91, 160] [20, 112, 122, 147] [1, 7, 35, 104] [28, 38, 63, 132] [48, 140, 150, 195] [68, 69, 87, 105] [96, 97, 115, 133] [9, 10, 31, 49] [40, 41, 59, 77] [152, 153, 182, 189] [124, 125, 143, 161] [14, 21, 168, 178] [71, 74, 106, 169] [89, 155, 158, 190] [33, 99, 102, 134] [43, 46, 78, 193] [12, 18, 50, 145] [22, 117, 180, 186] [61, 127, 130, 162]

H_Z (98 checks, sparse supports)

[41, 43, 77, 78] [10, 12, 49, 50] [153, 155, 189, 190] [97, 99, 133, 134] [21, 22, 178, 180] [125, 127, 161, 162] [69, 71, 105, 106] [123, 124, 136, 143] [95, 96, 108, 115] [39, 40, 52, 59] [13, 14, 164, 168] [67, 68, 80, 87] [8, 9, 24, 31] [151, 152, 181, 182] [128, 129, 131, 135] [184, 185, 187, 191] [16, 17, 19, 23] [44, 45, 47, 51] [100, 101, 103, 107] [156, 157, 159, 163] [72, 73, 75, 79] [175, 177, 192, 194] [81, 82, 83, 90] [53, 54, 55, 62] [109, 110, 111, 118] [0, 2, 165, 170] [25, 26, 27, 34] [137, 138, 139, 146] [33, 35, 102, 104] [18, 20, 145, 147] [117, 119, 186, 188] [46, 48, 193, 195] [61, 63, 130, 132] [89, 91, 158, 160] [74, 76, 169, 171] [4, 5, 11, 15] [86, 88, 92, 98] [58, 60, 64, 70] [172, 173, 179, 183] [114, 116, 120, 126] [142, 144, 148, 154] [30, 32, 36, 42] [84, 85, 93, 94] [140, 141, 149, 150] [166, 167, 174, 176] [1, 3, 6, 7] [56, 57, 65, 66] [112, 113, 121, 122] [28, 29, 37, 38] [50, 51, 145, 146] [106, 107, 169, 170] [61, 62, 162, 163] [78, 79, 193, 194] [33, 34, 134, 135] [89, 90, 190, 191] [22, 23, 117, 118] [122, 123, 137, 147] [164, 165, 166, 171] [7, 8, 25, 35] [38, 39, 53, 63] [94, 95, 109, 119] [150, 151, 192, 195] [66, 67, 81, 91] [126, 127, 129, 130] [70, 71, 73, 74] [98, 99, 101, 102] [179, 180, 185, 186] [154, 155, 157, 158] [42, 43, 45, 46] [11, 12, 17, 18] [120, 121, 124, 125] [92, 93, 96, 97] [36, 37, 40, 41] [167, 168, 173, 178] [64, 65, 68, 69] [5, 6, 9, 10] [148, 149, 152, 153] [31, 32, 44, 49] [143, 144, 156, 161] [115, 116, 128, 133] [182, 183, 184, 189] [59, 60, 72, 77] [87, 88, 100, 105] [14, 15, 16, 21] [27, 28, 131, 132] [55, 56, 159, 160] [75, 76, 175, 176] [0, 1, 103, 104] [19, 20, 111, 112] [83, 84, 187, 188] [47, 48, 139, 140] [24, 26, 29, 30] [108, 110, 113, 114] [52, 54, 57, 58] [2, 3, 4, 13] [172, 174, 177, 181] [136, 138, 141, 142] [80, 82, 85, 86]

Code ID 196-2-14-b · download JSON · raw on GitHub