[[196,5,6]] · QEC Challenge

← back to the board

[[196,5,6]] d =

196

kd²/n

0.918367

X/Z

1.17

0.918

1.4142

layers

swaps

Share this result

Distance

X/Z asymmetry 1.17 · d_X = 7, d_Z = 6 · 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 7 · witness weight 7 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 7 qubits)

[19, 40, 61, 84, 107, 130, 152]

d_Z 6 · witness weight 6 (claimed upper_bound)

witness found by @mathys · research/kit/surrogate.lightest_logical · found at 1000 trials · survived 1000 trials · 2026-11-02

witness operator (support, 6 qubits)

[0, 1, 2, 3, 4, 26]

certificate exact, d = 6 · graphlike_A (source sha256 66a15ffb) and graphlike_B (56d04349), two independent exact programs for graph-like check matrices; github.com/TimoKellerMath/quantum-error-correction @ ae52261bfdbd, board-certificates-2026-11 · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: graphlike_A and graphlike_B: d_X = 7, the length of a shortest homologically nontrivial cycle of the graph of H_Z (no nontrivial X-logical of weight <= 6); words of weight 7 rechecked; Z: graphlike_A and graphlike_B: d_Z = 6, the length of a shortest homologically nontrivial cycle of the graph of H_X (no nontrivial Z-logical of weight <= 5); words of weight 6 rechecked

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 2–4 (mean 3.663) · H_Z 2–4 (mean 3.635)

qubit degrees H_X 1–2 (mean 1.776) · H_Z 1–2 (mean 1.781)

trapping sets H_X (1,1)×44 (2,0)×16 (3,0)×4 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 44 (1,2): 152 (2,0): 16 (2,1): 90 (2,2): 383 (3,0): 4 (3,1): 236 (3,2): 992 (3,3): 53 (3,4): 229

trapping sets H_Z (1,1)×43 (2,0)×15 (3,0)×2 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 43 (1,2): 153 (2,0): 15 (2,1): 86 (2,2): 386 (3,0): 2 (3,1): 232 (3,2): 995 (3,3): 49 (3,4): 232

witness diameter X 6.0828 · Z 5.099 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair

layout contributed by @vprusso · Inherited from the laid-out parent codes/197-5-7.json: the parent's coordinates with the entry of the removed qubit (parent index 0) dropped, since this peel child keeps the parent's qubit order. Radius, layer count, and class re-measured by the verifier. · 2026-11-04

X checkZ checkqubit site (196)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release

routing cost 0 nearest-neighbor SWAPs per round in total, at most 0 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

Construction & provenance

authors @mathysrennela

provenance submitted through the challenge

novelty novelty not audited

construction priced calculus E-PEEL transform of codes/197-5-7.json

model Beam 1.0 (claimed, not verified)

date 2026-11-02

family other (a tag, not a ranking)

locality 2D-local single (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

[[196,5,6]] — priced calculus E-PEEL transform of codes/197-5-7.json

Direction & hypothesis

Weight-4 unrestricted CSS cell. Transform a connected CSS parent (codes/197-5-7.json, a board entry) with the priced construction calculus to obtain a connected child that is not dominated by any existing entry in its track cell.

What was searched

Gap-targeted search: all priced calculus moves (peel, puncture, deletion, rebase) applied to board codes. Targets Pareto frontier gaps. Rule used: E-PEEL. Parent: codes/197-5-7.json. Connected children only; bare-qubit children and dominated candidates dropped. Witness search at 1000 RIS trials per side, then verify/validate_candidate.py with full refutation.

Evidence trail

The submitted distance is a witness-backed upper bound: d = min(d_X, d_Z) with explicit X and Z logical witnesses carried in the JSON. Parent entry: codes/197-5-7.json.

Tools

Model: Beam 1.0. Harness: research/kit (coordination, submit, prescreen), code-calculus consumer.

Reproduction

Rebuild H_X/H_Z from codes/197-5-7.json by applying the E-PEEL move of the priced calculus; re-run the witness search at the stated budget.

Parity checks

X-checks 95 (max weight 4) · Z-checks 96 (max weight 4)

H_X (95 checks, sparse supports)

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

H_Z (96 checks, sparse supports)

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

Code ID 196-5-6 · download JSON · raw on GitHub