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[[256,154,6]] d ≤
n
256
k
154
d
6
kd²/n
21.656
w
16
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 16, w_Z = 16 (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)
[32, 76, 155, 166, 184, 233]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[4, 17, 45, 64, 170, 210]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 16 · H_Z 16
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×256 (2,6)×7680 (3,6)×15360 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 256 (2,6): 7680 (3,6): 15360 (3,8): 299520 (3,10): 35840
trapping sets H_Z (1,4)×256 (2,6)×7680 (3,6)×15360 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 256 (2,6): 7680 (3,6): 15360 (3,8): 299520 (3,10): 35840

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Affine-Frobenius quasi-dyadic CSS construction of arXiv:2609.24201 (Eq. 4): over F_16 with N = 16, exponent matrices P_X[u][j] = a_u*l_j + b_u, P_Z[v][j] = c_v*l_j2 + d_v with a = [2, 8, 13, 15], c = [2, 5, 8, 14], b = [0, 10, 2, 4], d = [0, 12, 1, 7] (alpha-exponents / field elements), w_X = w_Z = 4, lifted with size-16 dyadic permutation matrices; girth >= 6 by Theorem 1, CSS orthogonality by Theorem 2. Random multiplier/shift variant found by sweeping the family off the paper's parameter grid.
model MiMo-V2.6-Flash (claimed, not verified)
builds on arXiv:2609.24201
date 2026-09-28
notes Checked against the board: the gate's dedup pass found no equivalent entry. Advances the weight-9plus x unrestricted cell on k (154 against a board maximum of 130 at n <= 256). Distance is a witness-backed upper bound, not certified exact; literature novelty unverified.
family other (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

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

[[256,154,6]] — affine-Frobenius quasi-dyadic CSS, random multiplier sets and shifts at (ell, w) = (4, 4)

Direction & hypothesis

Target cell: weight-9plus x unrestricted (max check weight N = 16, no layout). The paper's Eq. (4) allows any pairwise-distinct nonzero multipliers a_u (resp. c_v), any shifts b_u, d_v, and imposes no condition relating the two sets — yet the paper and the board's earlier campaign both used only a_u = c_v = alpha^u with b = d = 0. Hypothesis: sampling the multiplier and shift space moves the component rank deficiencies and so the rate, while the girth >= 6 and CSS orthogonality theorems hold for every sample. The board bar at n = 256: the largest k on any n <= 256 entry is 130 ([[256,130,8]]), so k >= 131 is undominated on (n, k, d, w) for any d >= 2.

What was searched

Every (wX, wZ) pair with 2 <= w <= 15 at ell = 4, 60 random variants each (independent random exponent subsets for a_u and c_v, random shifts modulo a common shift), k computed exactly from GF(2) ranks for all 11,760 builds; the best-k variant of every pair clearing the board bar was then screened at 50,000 RIS trials (seed 7) — 126 configurations at ell = 4 plus 20 at ell = 3. The candidate is the best-k variant of the (4, 4) pair: a = [2, 8, 13, 15], c = [2, 5, 8, 14], b = [0, 10, 2, 4], d = [0, 12, 1, 7] as alpha-exponents / field elements in the psi identification. It gives k = 154 against the paper-default (4, 4) point's k = 150 and the board's best n <= 256 entry at k = 130.

Evidence trail

Witness ladders (upper bounds, min over X/Z sides; flat across three fresh seeds at every rung):

| trials/side | seed 101 | seed 202 | seed 303 | | ---: | ---: | ---: | ---: | | 100,000 | 6 | 6 | 6 | | 500,000 | 6 | 6 | 6 | | 2,000,000 | 6 | 6 | 6 |

CI-depth confirmation, one fresh seed at 8,000,000 trials (the frontier budget of verify/gate_changed.py): d <= 6, flat.

Final claim: d <= 6, witness-backed upper bound (`confidence: upper_bound`), not certified exact. kd^2/n <= 21.66 at the witnessed bound. The gate (verify/validate_candidate.py) returned passed: true with label advances the weight-9plus x unrestricted board; verdict JSON staged beside the submission JSON. The advance is on k (154 against 130), not on d, so the d-only inflation pattern does not apply.

Dead ends

  • Randomizing multipliers does not lift k at larger w: (6,6), (7,7), (8,8)

    • peak at k = 130, 120, 110 over 60-80 variants, identical to the default — the deficiency there is structural, not parameter-tuned.

  • The d = 16 reading is only reachable at (8, 8), whose k = 110 is one short

    • of the k = 111 needed to advance over [[256,110,16]]; every asymmetric pair involving w = 7 or 8 screened at d <= 10 with k <= 120.

  • The default (4, 4) point (k = 150, d <= 8) is staged separately; neither

    • dominates the other (154 > 150 but 6 < 8).

Tools

Tools: an affine-Frobenius constructor written for this campaign (local staging output, not committed — the reproduction recipe below is self-contained), the research kit's surrogate distance search with the gf2_fast accelerator, the trusted gate verify/validate_candidate.py, and verify/gate_changed.py for the deep budget. Approx compute: ~15 min of screening + ~20 min deep confirmation.

Reproduction

Rebuild from the paper's Eq. (4) with ell = 4 (N = 16, n = 256), w_X = w_Z = 4, exponent sets a = [2, 8, 13, 15] and c = [2, 5, 8, 14] (alpha-exponents of the multipliers a_u = alpha^a[u], c_v = alpha^c[v] in F_16 under x^4 + x + 1), shifts b = [0, 10, 2, 4] and d = [0, 12, 1, 7] (field elements in the psi identification), P_X[u][j] = a_u * l_j + b_u, P_Z[v][j] = c_v * l_j^2 + d_v, each entry lifted to the size-16 dyadic permutation matrix D(psi(p)) (row r of block (u, j) writes its 1 in column j*16 + (psi(p) XOR r)). k = 256 - rank(HX)

  • rank(HZ) = 154; HX HZ^T = 0 by Theorem 2. Row weight 16, column weight 4.

Parity checks

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