← back to the board
[[256,150,8]] d ≤
n
256
k
150
d
8
kd²/n
37.5
w
16
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[8, 25, 106, 123, 171, 186, 201, 216]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[31, 60, 67, 96, 143, 172, 211, 240]
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, P_Z[v][j] = c_v*l_j2 with a = c = alpha^u for u = 0..3, b = d = 0, w_X = w_Z = 4, lifted with size-16 dyadic permutation matrices; girth >= 6 by Theorem 1, CSS orthogonality by Theorem 2. Paper-default (ell, w) = (4, 4) instance, held back by an earlier kd2/n-ranked sweep and surfaced by re-ranking on the board's Pareto test.
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 (150 against a board maximum of 130 at n <= 256) at equal claimed distance d = 8. 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

[[256,150,8]] — affine-Frobenius quasi-dyadic CSS, paper-default multipliers at (ell, w) = (4, 4)

Direction & hypothesis

Target cell: weight-9plus x unrestricted (max check weight N = 16, no layout). The construction is the affine-Frobenius quasi-dyadic (QD) CSS family of arXiv:2609.24201 (Eq. 4), which guarantees CSS orthogonality and component girth >= 6 for every admissible parameter choice. The board's frontier in this cell at n = 256 was [[256,130,8]] (k = 130), [[256,120,10]] and [[256,110,16]], all from the same family's earlier campaign, so the opening was one axis wide: at n <= 256 the largest k on the board is 130, hence any candidate with k >= 131 is undominated on (n, k, d, w) regardless of where its distance lands above 2. Hypothesis: the paper only tabulates (ell, w) in {3,4} x {4,6,7,15}; intermediate w at ell = 4 should give k > 130 with the structural d = 8 the family reaches for w >= 4.

What was searched

Two sweeps of the family (n = N^2 with N = 2^ell, so only n = 64 and n = 256 are admissible under the blocklength rule: n = 1024 would need check weight 32 and exceeds the n <= 700 cap for w > 8):

  • an earlier sweep of 71 configurations (equal and |wX - wZ| <= 2 pairs, two

    • multiplier variants, b = d = 0, ell in {3,4}), screened at 50,000 RIS trials;

  • this campaign's wider sweep: every (wX, wZ) pair with 2 <= w <= N-1 at

    • ell in {3,4}, 60 random variants each (13,920 builds, k computed exactly for each from GF(2) ranks of HX and HZ), then the best-k variant of every pair whose k cleared the board bar screened at 50,000 RIS trials, seed 7 — 146 screened configurations in total (20 at ell = 3, 126 at ell = 4). Variants draw the multiplier exponents a_u, c_v as independent random subsets of the nonzero field elements and the shifts b_u, d_v randomly modulo a common shift; the paper fixes a_u = c_v = alpha^u, b_u = d_v = 0.

The candidate below is the paper-default (4, 4) point itself: k = 150 (component rank deficiencies 64 - 53 per side) against the paper's own (w = 6) k = 130. It was not on the earlier sweep's submission list because that campaign ranked by kd^2/n, where w = 8/8 scored higher; the board ranks on the Pareto frontier over (n, k, d, w), where k = 150 > 130 at equal d and w is a record.

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 | 8 | 8 | 8 | | 500,000 | 8 | 8 | 8 | | 2,000,000 | 8 | 8 | 8 |

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

Final claim: d <= 8, witness-backed upper bound (`confidence: upper_bound`), not certified exact. kd^2/n <= 37.5 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.

Dead ends

  • w = 6/6 and w = 5/7 at ell = 4 screen at k = 130, d <= 8 — same parameters

    • as the existing [[256,130,8]] board entry, so no axis is gained; not surfaced.

  • w >= 7 pairs reach d <= 10 only at k <= 120 ([[256,120,10]] already holds

    • that point); w = 8/8 variants never exceed k = 110 over 80 samples, one short of the k = 111 needed to advance over [[256,110,16]] at d = 16.

  • w <= 3 collapses to d <= 4 (high k, weak distance); ell = 5 (n = 1024) is

    • inadmissible under the blocklength rule.

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 reference. Screens ran on an M-series laptop; the deep confirmation passes took ~14-21 min each.

Reproduction

Build HX, HZ from the paper's Eq. (4) with ell = 4 (N = 16, n = 256), w_X = w_Z = 4, a_u = c_v = alpha^u for u = 0..3 (alpha a primitive element of F_16 under x^4 + x + 1), b_u = d_v = 0: exponent matrices P_X[u][j] = a_u * l_j and P_Z[v][j] = c_v * l_j^2 over F_16, each entry lifted to the dyadic permutation matrix D(psi(p)) of size 16, row r of block (u, j) writing its 1 in column j*16 + (psi(p) XOR r). k = 256 - rank(HX) - rank(HZ) = 150; HX HZ^T = 0 by the paper's 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, 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] [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] [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] [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] [0, 18, 36, 54, 72, 90, 108, 126, 131, 145, 167, 181, 203, 217, 239, 253] [1, 19, 37, 55, 73, 91, 109, 127, 130, 144, 166, 180, 202, 216, 238, 252] [2, 16, 38, 52, 74, 88, 110, 124, 129, 147, 165, 183, 201, 219, 237, 255] [3, 17, 39, 53, 75, 89, 111, 125, 128, 146, 164, 182, 200, 218, 236, 254] [4, 22, 32, 50, 76, 94, 104, 122, 135, 149, 163, 177, 207, 221, 235, 249] [5, 23, 33, 51, 77, 95, 105, 123, 134, 148, 162, 176, 206, 220, 234, 248] [6, 20, 34, 48, 78, 92, 106, 120, 133, 151, 161, 179, 205, 223, 233, 251] [7, 21, 35, 49, 79, 93, 107, 121, 132, 150, 160, 178, 204, 222, 232, 250] [8, 26, 44, 62, 64, 82, 100, 118, 139, 153, 175, 189, 195, 209, 231, 245] [9, 27, 45, 63, 65, 83, 101, 119, 138, 152, 174, 188, 194, 208, 230, 244] [10, 24, 46, 60, 66, 80, 102, 116, 137, 155, 173, 191, 193, 211, 229, 247] [11, 25, 47, 61, 67, 81, 103, 117, 136, 154, 172, 190, 192, 210, 228, 246] [12, 30, 40, 58, 68, 86, 96, 114, 143, 157, 171, 185, 199, 213, 227, 241] [13, 31, 41, 59, 69, 87, 97, 115, 142, 156, 170, 184, 198, 212, 226, 240] [14, 28, 42, 56, 70, 84, 98, 112, 141, 159, 169, 187, 197, 215, 225, 243] [15, 29, 43, 57, 71, 85, 99, 113, 140, 158, 168, 186, 196, 214, 224, 242] [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] [0, 24, 35, 59, 70, 94, 101, 125, 140, 148, 175, 183, 202, 210, 233, 241] [1, 25, 34, 58, 71, 95, 100, 124, 141, 149, 174, 182, 203, 211, 232, 240] [2, 26, 33, 57, 68, 92, 103, 127, 142, 150, 173, 181, 200, 208, 235, 243] [3, 27, 32, 56, 69, 93, 102, 126, 143, 151, 172, 180, 201, 209, 234, 242] [4, 28, 39, 63, 66, 90, 97, 121, 136, 144, 171, 179, 206, 214, 237, 245] [5, 29, 38, 62, 67, 91, 96, 120, 137, 145, 170, 178, 207, 215, 236, 244] [6, 30, 37, 61, 64, 88, 99, 123, 138, 146, 169, 177, 204, 212, 239, 247] [7, 31, 36, 60, 65, 89, 98, 122, 139, 147, 168, 176, 205, 213, 238, 246] [8, 16, 43, 51, 78, 86, 109, 117, 132, 156, 167, 191, 194, 218, 225, 249] [9, 17, 42, 50, 79, 87, 108, 116, 133, 157, 166, 190, 195, 219, 224, 248] [10, 18, 41, 49, 76, 84, 111, 119, 134, 158, 165, 189, 192, 216, 227, 251] [11, 19, 40, 48, 77, 85, 110, 118, 135, 159, 164, 188, 193, 217, 226, 250] [12, 20, 47, 55, 74, 82, 105, 113, 128, 152, 163, 187, 198, 222, 229, 253] [13, 21, 46, 54, 75, 83, 104, 112, 129, 153, 162, 186, 199, 223, 228, 252] [14, 22, 45, 53, 72, 80, 107, 115, 130, 154, 161, 185, 196, 220, 231, 255] [15, 23, 44, 52, 73, 81, 106, 114, 131, 155, 160, 184, 197, 221, 230, 254]
H_Z (64 checks, sparse supports)
[0, 17, 36, 53, 67, 82, 103, 118, 140, 157, 168, 185, 207, 222, 235, 250] [1, 16, 37, 52, 66, 83, 102, 119, 141, 156, 169, 184, 206, 223, 234, 251] [2, 19, 38, 55, 65, 80, 101, 116, 142, 159, 170, 187, 205, 220, 233, 248] [3, 18, 39, 54, 64, 81, 100, 117, 143, 158, 171, 186, 204, 221, 232, 249] [4, 21, 32, 49, 71, 86, 99, 114, 136, 153, 172, 189, 203, 218, 239, 254] [5, 20, 33, 48, 70, 87, 98, 115, 137, 152, 173, 188, 202, 219, 238, 255] [6, 23, 34, 51, 69, 84, 97, 112, 138, 155, 174, 191, 201, 216, 237, 252] [7, 22, 35, 50, 68, 85, 96, 113, 139, 154, 175, 190, 200, 217, 236, 253] [8, 25, 44, 61, 75, 90, 111, 126, 132, 149, 160, 177, 199, 214, 227, 242] [9, 24, 45, 60, 74, 91, 110, 127, 133, 148, 161, 176, 198, 215, 226, 243] [10, 27, 46, 63, 73, 88, 109, 124, 134, 151, 162, 179, 197, 212, 225, 240] [11, 26, 47, 62, 72, 89, 108, 125, 135, 150, 163, 178, 196, 213, 224, 241] [12, 29, 40, 57, 79, 94, 107, 122, 128, 145, 164, 181, 195, 210, 231, 246] [13, 28, 41, 56, 78, 95, 106, 123, 129, 144, 165, 180, 194, 211, 230, 247] [14, 31, 42, 59, 77, 92, 105, 120, 130, 147, 166, 183, 193, 208, 229, 244] [15, 30, 43, 58, 76, 93, 104, 121, 131, 146, 167, 182, 192, 209, 228, 245] [0, 18, 40, 58, 70, 84, 110, 124, 139, 153, 163, 177, 205, 223, 229, 247] [1, 19, 41, 59, 71, 85, 111, 125, 138, 152, 162, 176, 204, 222, 228, 246] [2, 16, 42, 56, 68, 86, 108, 126, 137, 155, 161, 179, 207, 221, 231, 245] [3, 17, 43, 57, 69, 87, 109, 127, 136, 154, 160, 178, 206, 220, 230, 244] [4, 22, 44, 62, 66, 80, 106, 120, 143, 157, 167, 181, 201, 219, 225, 243] [5, 23, 45, 63, 67, 81, 107, 121, 142, 156, 166, 180, 200, 218, 224, 242] [6, 20, 46, 60, 64, 82, 104, 122, 141, 159, 165, 183, 203, 217, 227, 241] [7, 21, 47, 61, 65, 83, 105, 123, 140, 158, 164, 182, 202, 216, 226, 240] [8, 26, 32, 50, 78, 92, 102, 116, 131, 145, 171, 185, 197, 215, 237, 255] [9, 27, 33, 51, 79, 93, 103, 117, 130, 144, 170, 184, 196, 214, 236, 254] [10, 24, 34, 48, 76, 94, 100, 118, 129, 147, 169, 187, 199, 213, 239, 253] [11, 25, 35, 49, 77, 95, 101, 119, 128, 146, 168, 186, 198, 212, 238, 252] [12, 30, 36, 54, 74, 88, 98, 112, 135, 149, 175, 189, 193, 211, 233, 251] [13, 31, 37, 55, 75, 89, 99, 113, 134, 148, 174, 188, 192, 210, 232, 250] [14, 28, 38, 52, 72, 90, 96, 114, 133, 151, 173, 191, 195, 209, 235, 249] [15, 29, 39, 53, 73, 91, 97, 115, 132, 150, 172, 190, 194, 208, 234, 248] [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] [0, 24, 38, 62, 75, 83, 109, 117, 138, 146, 172, 180, 193, 217, 231, 255] [1, 25, 39, 63, 74, 82, 108, 116, 139, 147, 173, 181, 192, 216, 230, 254] [2, 26, 36, 60, 73, 81, 111, 119, 136, 144, 174, 182, 195, 219, 229, 253] [3, 27, 37, 61, 72, 80, 110, 118, 137, 145, 175, 183, 194, 218, 228, 252] [4, 28, 34, 58, 79, 87, 105, 113, 142, 150, 168, 176, 197, 221, 227, 251] [5, 29, 35, 59, 78, 86, 104, 112, 143, 151, 169, 177, 196, 220, 226, 250] [6, 30, 32, 56, 77, 85, 107, 115, 140, 148, 170, 178, 199, 223, 225, 249] [7, 31, 33, 57, 76, 84, 106, 114, 141, 149, 171, 179, 198, 222, 224, 248] [8, 16, 46, 54, 67, 91, 101, 125, 130, 154, 164, 188, 201, 209, 239, 247] [9, 17, 47, 55, 66, 90, 100, 124, 131, 155, 165, 189, 200, 208, 238, 246] [10, 18, 44, 52, 65, 89, 103, 127, 128, 152, 166, 190, 203, 211, 237, 245] [11, 19, 45, 53, 64, 88, 102, 126, 129, 153, 167, 191, 202, 210, 236, 244] [12, 20, 42, 50, 71, 95, 97, 121, 134, 158, 160, 184, 205, 213, 235, 243] [13, 21, 43, 51, 70, 94, 96, 120, 135, 159, 161, 185, 204, 212, 234, 242] [14, 22, 40, 48, 69, 93, 99, 123, 132, 156, 162, 186, 207, 215, 233, 241] [15, 23, 41, 49, 68, 92, 98, 122, 133, 157, 163, 187, 206, 214, 232, 240]
Code ID 256-150-8 · download JSON · raw on GitHub