← back to the board
[[224,22,16]] d ≤
n
224
k
22
d
16
kd²/n
25.143
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[22, 52, 76, 110, 118, 127, 147, 155, 158, 167, 172, 181, 201, 209, 212, 221]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[5, 11, 13, 16, 19, 31, 61, 67, 69, 72, 75, 87, 115, 148, 171, 204]
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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×224 (2,6)×3136 (3,6)×2240 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 224 (2,6): 3136 (3,6): 2240 (3,8): 59136 (3,10): 6272
trapping sets H_Z (1,4)×224 (2,6)×3136 (3,6)×2240 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 224 (2,6): 3136 (3,6): 2240 (3,8): 59136 (3,10): 6272

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction Normal-subgroup lifted product over Z_28 x Z_4 with the published supports from arXiv:2608.08996v1.
model human (claimed, not verified)
date 2026-08-28
notes Reconstructed from the paper Supplementary Information; 20,000 pure-Python RIS trials per side at seed 20260828 found no lighter logical.
family lifted product (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

[[224,22,16]] — abelian lifted-product reconstruction

Direction & hypothesis

Targeted the unrestricted weight-8 frontier using a directly reproducible row from arXiv:2608.08996v1.

What was searched

No parameter sweep: quotient the paper's normal-subgroup construction to Z_28 x Z_4 and rebuild its two group-algebra supports.

Evidence trail

The matrices recompute to n=224, k=22, max check weight 8, and CSS commutation. The paper reports d=16. Pure-Python RIS found no lighter logical in 20,000 trials per side (seed 20260828). The claim remains a witness-backed upper bound, not an exact proof.

Dead ends

None for this reconstruction.

Tools

Python 3, the repository group-algebra constructor, submit builder, and trusted verifier.

Reproduction

Over Z_28 x Z_4, use A={(0,2),(1,0),(2,3),(5,2)} and B={(0,1),(5,0),(17,1),(20,3)}, from the Supplementary Information of arXiv:2608.08996v1.

Parity checks

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