[[324,196,4]] · QEC Challenge

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · w_X = 30, w_Z = 30 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[184, 191, 238, 245]

d_Z 4 · witness weight 4 (claimed upper_bound)

witness operator (support, 4 qubits)

[184, 191, 238, 245]

certificate exact, d = 4 · tropidist b03ec2d4077f (arXiv:2611.08247) exhaustive search · level solver: a solver verdict, no artifact a third party can check without re-running the encoding

X: no logical < 4 exists; Z: no logical < 4 exists

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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 4–30 (mean 18.75) · H_Z 4–30 (mean 18.75)

qubit degrees H_X 1–30 (mean 3.704) · H_Z 1–30 (mean 3.704)

trapping sets H_X (1,1)×18 (2,1)×36 (3,1)×694 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 18 (1,2): 242 (1,3): 28 (1,15): 16 (1,16): 16 (1,29): 2 (1,30): 2 (2,1): 36 (2,2): 2535 (2,3): 590 (2,4): 239 (2,14): 256 (2,15): 1824 (2,16): 2048 (2,17): 512 (2,18): 32 (2,27): 28 (2,28): 60 (2,29): 480 (2,30): 448 (3,1): 694 (3,2): 22510 (3,3): 12816 (3,4): 23848 (3,5): 15000 (3,6): 8402 (3,7): 2296 (3,13): 12096 (3,14): 35616 (3,15): 96016 (3,16): 123168 (3,17): 64016 (3,18): 20768 (3,19): 10688 (3,20): 3264 (3,25): 364 (3,26): 784 (3,27): 9212 (3,28): 15540 (3,29): 54910 (3,30): 50464 (3,31): 3982 (3,32): 1904 (3,33): 336 (3,34): 112

trapping sets H_Z (1,1)×18 (2,1)×36 (3,1)×694 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,1): 18 (1,2): 242 (1,3): 28 (1,15): 16 (1,16): 16 (1,29): 2 (1,30): 2 (2,1): 36 (2,2): 2535 (2,3): 590 (2,4): 239 (2,14): 256 (2,15): 1824 (2,16): 2048 (2,17): 512 (2,18): 32 (2,27): 28 (2,28): 60 (2,29): 480 (2,30): 448 (3,1): 694 (3,2): 22510 (3,3): 12816 (3,4): 23848 (3,5): 15000 (3,6): 8402 (3,7): 2296 (3,13): 12096 (3,14): 35616 (3,15): 96016 (3,16): 123168 (3,17): 64016 (3,18): 20768 (3,19): 10688 (3,20): 3264 (3,25): 364 (3,26): 784 (3,27): 9212 (3,28): 15540 (3,29): 54910 (3,30): 50464 (3,31): 3982 (3,32): 1904 (3,33): 336 (3,34): 112

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction H-code inner [[18,14,2]] plus sparse self-dual CSS outer [[18,14,2]] (checks {0..9},{8..17}); tree concatenation of arXiv:2610.30087 Appendix B.

model DeepSeek V4.1 Flash 1017 (claimed, not verified)

date 2026-10-25

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

[[324,196,4]] — H-code inner / sparse [[18,14,2]] outer tree concatenation

Direction & hypothesis

Cell: CSS, unrestricted, weight-9plus. The board has no CSS entry with n <= 324, k >= 196 and d >= 4. The H-code tree concatenation of arXiv:2610.30087 Appendix B gives [[(k+4)^2,k^2,4]], but the pure self-concatenation at k=14 has check weight 3(k+2)=48, above the verifier's 32-qubit cap. Hypothesis: keep the H code as the inner code and replace the outer H code with a sparse [[18,14,2]] CSS code with two weight-10 checks per side, reducing the lifted heavy check from 48 to 30.

What was searched

Reconstructed the H-code family for k=2,...,16. Pure self-concatenations are dominated at k <= 8 and inadmissible at k >= 10. For k0=14, an outer self-dual CSS code with two even checks of weight 10 covers all 18 outer qubits; lower even maximum is impossible for a connected two-check cover of 18 qubits. The resulting lifted weight is 3*10=30, under the 32 cap.

Evidence trail

inner H code [[18,14,2]]: checks A0={0,1,2,3} and B0={0,1,4,...,17};

logical representatives L_a={1,3,4+a}, a=0,...,13, weight 3.

outer self-dual CSS [[18,14,2]]: checks A={0,...,9} and B={8,...,17},

both weight 10, overlapping in two qubits, union all 18.

For inner block b and outer block a, qubit (b,q) is b*18+q; an outer check S lifts on outer block a to the union over b in S of {b*18+q : q in L_a}. H_X and H_Z are the inner checks on every block plus the lifted outer checks for every outer block.

The result has n=324, k=196, max check weight 30, CSS commutation, and a connected combined Tanner graph. Explicit weight-4 X and Z witnesses lie on support [0,1,18,19]. Exhaustive search found no nontrivial logical of weight 1, 2 or 3, and the board's 8000-trial RIS refutation found none lighter. The claim is a witness-backed upper bound d <= 4; the concatenation theorem gives d = 4.

Dead ends

Pure self-concatenated H codes: k <= 8 dominated, k >= 10 above the weight cap. The pure k=14 code [[324,196,4]] has w=48 and is inadmissible. A single check per side cannot cover 18 qubits under the cap; three checks on one side would require a weight-18 cover. The two-even-weight-check cover is minimal.

Tools

Model: DeepSeek V4.1 Flash 1017, via the OpenCode harness. Construction and GF(2) checks were written locally; distance was checked by exhaustive weight<=3 enumeration and the board verifier's refutation pass.

Reproduction

Use the inner and outer checks above with the tree-concatenation rule in the evidence trail. The construction is exact and seed-free.

Parity checks

X-checks 64 (max weight 30) · Z-checks 64 (max weight 30)

H_X (64 checks, sparse supports)

[0, 1, 2, 3] [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] [18, 19, 20, 21] [18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] [36, 37, 38, 39] [36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] [54, 55, 56, 57] [54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] [72, 73, 74, 75] [72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89] [90, 91, 92, 93] [90, 91, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107] [108, 109, 110, 111] [108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] [126, 127, 128, 129] [126, 127, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143] [144, 145, 146, 147] [144, 145, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161] [162, 163, 164, 165] [162, 163, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179] [180, 181, 182, 183] [180, 181, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197] [198, 199, 200, 201] [198, 199, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215] [216, 217, 218, 219] [216, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233] [234, 235, 236, 237] [234, 235, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251] [252, 253, 254, 255] [252, 253, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269] [270, 271, 272, 273] [270, 271, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287] [288, 289, 290, 291] [288, 289, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305] [306, 307, 308, 309] [306, 307, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323] [1, 3, 4, 19, 21, 22, 37, 39, 40, 55, 57, 58, 73, 75, 76, 91, 93, 94, 109, 111, 112, 127, 129, 130, 145, 147, 148, 163, 165, 166] [145, 147, 148, 163, 165, 166, 181, 183, 184, 199, 201, 202, 217, 219, 220, 235, 237, 238, 253, 255, 256, 271, 273, 274, 289, 291, 292, 307, 309, 310] [1, 3, 5, 19, 21, 23, 37, 39, 41, 55, 57, 59, 73, 75, 77, 91, 93, 95, 109, 111, 113, 127, 129, 131, 145, 147, 149, 163, 165, 167] [145, 147, 149, 163, 165, 167, 181, 183, 185, 199, 201, 203, 217, 219, 221, 235, 237, 239, 253, 255, 257, 271, 273, 275, 289, 291, 293, 307, 309, 311] [1, 3, 6, 19, 21, 24, 37, 39, 42, 55, 57, 60, 73, 75, 78, 91, 93, 96, 109, 111, 114, 127, 129, 132, 145, 147, 150, 163, 165, 168] [145, 147, 150, 163, 165, 168, 181, 183, 186, 199, 201, 204, 217, 219, 222, 235, 237, 240, 253, 255, 258, 271, 273, 276, 289, 291, 294, 307, 309, 312] [1, 3, 7, 19, 21, 25, 37, 39, 43, 55, 57, 61, 73, 75, 79, 91, 93, 97, 109, 111, 115, 127, 129, 133, 145, 147, 151, 163, 165, 169] [145, 147, 151, 163, 165, 169, 181, 183, 187, 199, 201, 205, 217, 219, 223, 235, 237, 241, 253, 255, 259, 271, 273, 277, 289, 291, 295, 307, 309, 313] [1, 3, 8, 19, 21, 26, 37, 39, 44, 55, 57, 62, 73, 75, 80, 91, 93, 98, 109, 111, 116, 127, 129, 134, 145, 147, 152, 163, 165, 170] [145, 147, 152, 163, 165, 170, 181, 183, 188, 199, 201, 206, 217, 219, 224, 235, 237, 242, 253, 255, 260, 271, 273, 278, 289, 291, 296, 307, 309, 314] [1, 3, 9, 19, 21, 27, 37, 39, 45, 55, 57, 63, 73, 75, 81, 91, 93, 99, 109, 111, 117, 127, 129, 135, 145, 147, 153, 163, 165, 171] [145, 147, 153, 163, 165, 171, 181, 183, 189, 199, 201, 207, 217, 219, 225, 235, 237, 243, 253, 255, 261, 271, 273, 279, 289, 291, 297, 307, 309, 315] [1, 3, 10, 19, 21, 28, 37, 39, 46, 55, 57, 64, 73, 75, 82, 91, 93, 100, 109, 111, 118, 127, 129, 136, 145, 147, 154, 163, 165, 172] [145, 147, 154, 163, 165, 172, 181, 183, 190, 199, 201, 208, 217, 219, 226, 235, 237, 244, 253, 255, 262, 271, 273, 280, 289, 291, 298, 307, 309, 316] [1, 3, 11, 19, 21, 29, 37, 39, 47, 55, 57, 65, 73, 75, 83, 91, 93, 101, 109, 111, 119, 127, 129, 137, 145, 147, 155, 163, 165, 173] [145, 147, 155, 163, 165, 173, 181, 183, 191, 199, 201, 209, 217, 219, 227, 235, 237, 245, 253, 255, 263, 271, 273, 281, 289, 291, 299, 307, 309, 317] [1, 3, 12, 19, 21, 30, 37, 39, 48, 55, 57, 66, 73, 75, 84, 91, 93, 102, 109, 111, 120, 127, 129, 138, 145, 147, 156, 163, 165, 174] [145, 147, 156, 163, 165, 174, 181, 183, 192, 199, 201, 210, 217, 219, 228, 235, 237, 246, 253, 255, 264, 271, 273, 282, 289, 291, 300, 307, 309, 318] [1, 3, 13, 19, 21, 31, 37, 39, 49, 55, 57, 67, 73, 75, 85, 91, 93, 103, 109, 111, 121, 127, 129, 139, 145, 147, 157, 163, 165, 175] [145, 147, 157, 163, 165, 175, 181, 183, 193, 199, 201, 211, 217, 219, 229, 235, 237, 247, 253, 255, 265, 271, 273, 283, 289, 291, 301, 307, 309, 319] [1, 3, 14, 19, 21, 32, 37, 39, 50, 55, 57, 68, 73, 75, 86, 91, 93, 104, 109, 111, 122, 127, 129, 140, 145, 147, 158, 163, 165, 176] [145, 147, 158, 163, 165, 176, 181, 183, 194, 199, 201, 212, 217, 219, 230, 235, 237, 248, 253, 255, 266, 271, 273, 284, 289, 291, 302, 307, 309, 320] [1, 3, 15, 19, 21, 33, 37, 39, 51, 55, 57, 69, 73, 75, 87, 91, 93, 105, 109, 111, 123, 127, 129, 141, 145, 147, 159, 163, 165, 177] [145, 147, 159, 163, 165, 177, 181, 183, 195, 199, 201, 213, 217, 219, 231, 235, 237, 249, 253, 255, 267, 271, 273, 285, 289, 291, 303, 307, 309, 321] [1, 3, 16, 19, 21, 34, 37, 39, 52, 55, 57, 70, 73, 75, 88, 91, 93, 106, 109, 111, 124, 127, 129, 142, 145, 147, 160, 163, 165, 178] [145, 147, 160, 163, 165, 178, 181, 183, 196, 199, 201, 214, 217, 219, 232, 235, 237, 250, 253, 255, 268, 271, 273, 286, 289, 291, 304, 307, 309, 322] [1, 3, 17, 19, 21, 35, 37, 39, 53, 55, 57, 71, 73, 75, 89, 91, 93, 107, 109, 111, 125, 127, 129, 143, 145, 147, 161, 163, 165, 179] [145, 147, 161, 163, 165, 179, 181, 183, 197, 199, 201, 215, 217, 219, 233, 235, 237, 251, 253, 255, 269, 271, 273, 287, 289, 291, 305, 307, 309, 323]

H_Z (64 checks, sparse supports)

[0, 1, 2, 3] [0, 1, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17] [18, 19, 20, 21] [18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] [36, 37, 38, 39] [36, 37, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53] [54, 55, 56, 57] [54, 55, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71] [72, 73, 74, 75] [72, 73, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89] [90, 91, 92, 93] [90, 91, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107] [108, 109, 110, 111] [108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] [126, 127, 128, 129] [126, 127, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143] [144, 145, 146, 147] [144, 145, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161] [162, 163, 164, 165] [162, 163, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179] [180, 181, 182, 183] [180, 181, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197] [198, 199, 200, 201] [198, 199, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215] [216, 217, 218, 219] [216, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233] [234, 235, 236, 237] [234, 235, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251] [252, 253, 254, 255] [252, 253, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269] [270, 271, 272, 273] [270, 271, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287] [288, 289, 290, 291] [288, 289, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305] [306, 307, 308, 309] [306, 307, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323] [1, 3, 4, 19, 21, 22, 37, 39, 40, 55, 57, 58, 73, 75, 76, 91, 93, 94, 109, 111, 112, 127, 129, 130, 145, 147, 148, 163, 165, 166] [145, 147, 148, 163, 165, 166, 181, 183, 184, 199, 201, 202, 217, 219, 220, 235, 237, 238, 253, 255, 256, 271, 273, 274, 289, 291, 292, 307, 309, 310] [1, 3, 5, 19, 21, 23, 37, 39, 41, 55, 57, 59, 73, 75, 77, 91, 93, 95, 109, 111, 113, 127, 129, 131, 145, 147, 149, 163, 165, 167] [145, 147, 149, 163, 165, 167, 181, 183, 185, 199, 201, 203, 217, 219, 221, 235, 237, 239, 253, 255, 257, 271, 273, 275, 289, 291, 293, 307, 309, 311] [1, 3, 6, 19, 21, 24, 37, 39, 42, 55, 57, 60, 73, 75, 78, 91, 93, 96, 109, 111, 114, 127, 129, 132, 145, 147, 150, 163, 165, 168] [145, 147, 150, 163, 165, 168, 181, 183, 186, 199, 201, 204, 217, 219, 222, 235, 237, 240, 253, 255, 258, 271, 273, 276, 289, 291, 294, 307, 309, 312] [1, 3, 7, 19, 21, 25, 37, 39, 43, 55, 57, 61, 73, 75, 79, 91, 93, 97, 109, 111, 115, 127, 129, 133, 145, 147, 151, 163, 165, 169] [145, 147, 151, 163, 165, 169, 181, 183, 187, 199, 201, 205, 217, 219, 223, 235, 237, 241, 253, 255, 259, 271, 273, 277, 289, 291, 295, 307, 309, 313] [1, 3, 8, 19, 21, 26, 37, 39, 44, 55, 57, 62, 73, 75, 80, 91, 93, 98, 109, 111, 116, 127, 129, 134, 145, 147, 152, 163, 165, 170] [145, 147, 152, 163, 165, 170, 181, 183, 188, 199, 201, 206, 217, 219, 224, 235, 237, 242, 253, 255, 260, 271, 273, 278, 289, 291, 296, 307, 309, 314] [1, 3, 9, 19, 21, 27, 37, 39, 45, 55, 57, 63, 73, 75, 81, 91, 93, 99, 109, 111, 117, 127, 129, 135, 145, 147, 153, 163, 165, 171] [145, 147, 153, 163, 165, 171, 181, 183, 189, 199, 201, 207, 217, 219, 225, 235, 237, 243, 253, 255, 261, 271, 273, 279, 289, 291, 297, 307, 309, 315] [1, 3, 10, 19, 21, 28, 37, 39, 46, 55, 57, 64, 73, 75, 82, 91, 93, 100, 109, 111, 118, 127, 129, 136, 145, 147, 154, 163, 165, 172] [145, 147, 154, 163, 165, 172, 181, 183, 190, 199, 201, 208, 217, 219, 226, 235, 237, 244, 253, 255, 262, 271, 273, 280, 289, 291, 298, 307, 309, 316] [1, 3, 11, 19, 21, 29, 37, 39, 47, 55, 57, 65, 73, 75, 83, 91, 93, 101, 109, 111, 119, 127, 129, 137, 145, 147, 155, 163, 165, 173] [145, 147, 155, 163, 165, 173, 181, 183, 191, 199, 201, 209, 217, 219, 227, 235, 237, 245, 253, 255, 263, 271, 273, 281, 289, 291, 299, 307, 309, 317] [1, 3, 12, 19, 21, 30, 37, 39, 48, 55, 57, 66, 73, 75, 84, 91, 93, 102, 109, 111, 120, 127, 129, 138, 145, 147, 156, 163, 165, 174] [145, 147, 156, 163, 165, 174, 181, 183, 192, 199, 201, 210, 217, 219, 228, 235, 237, 246, 253, 255, 264, 271, 273, 282, 289, 291, 300, 307, 309, 318] [1, 3, 13, 19, 21, 31, 37, 39, 49, 55, 57, 67, 73, 75, 85, 91, 93, 103, 109, 111, 121, 127, 129, 139, 145, 147, 157, 163, 165, 175] [145, 147, 157, 163, 165, 175, 181, 183, 193, 199, 201, 211, 217, 219, 229, 235, 237, 247, 253, 255, 265, 271, 273, 283, 289, 291, 301, 307, 309, 319] [1, 3, 14, 19, 21, 32, 37, 39, 50, 55, 57, 68, 73, 75, 86, 91, 93, 104, 109, 111, 122, 127, 129, 140, 145, 147, 158, 163, 165, 176] [145, 147, 158, 163, 165, 176, 181, 183, 194, 199, 201, 212, 217, 219, 230, 235, 237, 248, 253, 255, 266, 271, 273, 284, 289, 291, 302, 307, 309, 320] [1, 3, 15, 19, 21, 33, 37, 39, 51, 55, 57, 69, 73, 75, 87, 91, 93, 105, 109, 111, 123, 127, 129, 141, 145, 147, 159, 163, 165, 177] [145, 147, 159, 163, 165, 177, 181, 183, 195, 199, 201, 213, 217, 219, 231, 235, 237, 249, 253, 255, 267, 271, 273, 285, 289, 291, 303, 307, 309, 321] [1, 3, 16, 19, 21, 34, 37, 39, 52, 55, 57, 70, 73, 75, 88, 91, 93, 106, 109, 111, 124, 127, 129, 142, 145, 147, 160, 163, 165, 178] [145, 147, 160, 163, 165, 178, 181, 183, 196, 199, 201, 214, 217, 219, 232, 235, 237, 250, 253, 255, 268, 271, 273, 286, 289, 291, 304, 307, 309, 322] [1, 3, 17, 19, 21, 35, 37, 39, 53, 55, 57, 71, 73, 75, 89, 91, 93, 107, 109, 111, 125, 127, 129, 143, 145, 147, 161, 163, 165, 179] [145, 147, 161, 163, 165, 179, 181, 183, 197, 199, 201, 215, 217, 219, 233, 235, 237, 251, 253, 255, 269, 271, 273, 287, 289, 291, 305, 307, 309, 323]

Code ID 324-196-4 · download JSON · raw on GitHub