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[[256,6,6]] d ≤
n
256
k
6
d
6
kd²/n
0.844
w
4
X/Z
1
g
0.844
r
1.4142
layers
1
swaps
0

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Distance

X/Z asymmetry 1 · d_X ≤ 6, 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[3, 4, 5, 15, 27, 28]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[121, 122, 123, 126, 143, 144]
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 2–4 (mean 3.632) · H_Z 2–4 (mean 3.64)
qubit degrees H_X 1–2 (mean 1.773) · H_Z 1–2 (mean 1.777)
trapping sets H_X (1,1)×58 (2,0)×21 (3,0)×3 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,1): 58 (1,2): 198 (2,0): 21 (2,1): 119 (2,2): 493 (3,0): 3 (3,1): 329 (3,2): 1245 (3,3): 70 (3,4): 292
trapping sets H_Z (1,1)×57 (2,0)×21 (3,0)×5 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,1): 57 (1,2): 199 (2,0): 21 (2,1): 119 (2,2): 496 (3,0): 5 (3,1): 321 (3,2): 1258 (3,3): 72 (3,4): 293
witness diameter X 5.3852 · Z 5.0 (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
r = 1.414
X checkZ checkqubit site (256)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 Two degree-one r=1 grafts of the [[258,6,7]] multi-band dense-packed code
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-17
notes Checked against the board: no exact-duplicate or WL-equivalent entry (trusted gate).
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

[[256,6,6]] — two r=1 grafts of the [[258,6,7]] multi-band dense-packed code

Direction & hypothesis

Issue #1155 hackathon, 2D-local priority. Two degree-one r=1 grafts remove two qubits at held (k, d), shaving n on a current single-layer local-frontier source.

What was searched

Source: [[258,6,7]] (multi-band dense-packed code, single-layer 2D-local layout). Move: two degree-one r=1 grafts. Original column identities tracked so the source coordinates subset honestly; layers preserved.

Evidence trail

Trusted gate (verify/validate_candidate, refute=True): passed=true, board_advancing=true in the weight-4 × local-2d-single cell, no exact duplicate / WL-equivalent board entry, not refuted. Fresh witness search at submission: d ≤ 6, upper bound. Not an exact certificate; literature novelty unverified.

Dead ends

Two high-k single-layer leads ([[453,8,18]], [[452,8,17]]) were refuted at the gate and excluded. Plain deletion-control and incident-deletion-control moves produced no board-advancing survivors on their own.

Tools

Model: DeepSeek V4 Flash 0731 (Zed agent). Repo tooling: research/kit (make_submission, css GF(2) exports), verify/validate_candidate (trusted gate), verify/qldpc_verify (structural screening + board reports). Local compute only.

Reproduction

Rebuild (H_X, H_Z) from the source [[258,6,7]] multi-band dense-packed matrix: 1. Graft two degree-one qubits in sequence: for each, pick a column q that participates in exactly one stabilizer of some Pauli type, remove that column and its unique same-side row, and truncate opposite-type rows containing q (commutation is preserved by construction). 2. Subset the source's 2D coordinates to the surviving columns (original identities, not guessed) and keep the single-layer layout.

The method is the r=1 lattice grafting of arXiv:2504.08887 Sec. III E; the exact deleted row/column identities are deterministic from the source and the move order described above. The packaged JSON carries embedded witnesses and seed for the distance claim.

Parity checks

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