{ "schema_version": "0.2", "name": "[[225,104,3]] reduction of [[232,104,3]]", "code_type": "CSS", "n": 225, "k": 104, "checks": { "X": [ [ 0, 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196, 210, 217 ], [ 1, 15, 29, 43, 57, 71, 85, 99, 113, 127, 141, 155, 169, 183, 197, 211 ], [ 2, 16, 30, 44, 58, 72, 86, 100, 114, 128, 142, 156, 170, 184, 198, 212, 217 ], [ 3, 17, 31, 45, 59, 73, 87, 101, 115, 129, 143, 157, 171, 185, 199, 213, 218 ], [ 4, 18, 32, 46, 60, 74, 88, 102, 116, 130, 144, 158, 172, 186, 200, 214, 217, 218 ], [ 5, 19, 33, 47, 61, 75, 89, 103, 117, 131, 145, 159, 173, 187, 201, 215, 218 ], [ 6, 20, 34, 48, 62, 76, 90, 104, 118, 132, 146, 160, 174, 188, 202, 216, 217, 218 ], [ 7, 14, 35, 42, 63, 70, 91, 98, 119, 126, 147, 154, 175, 182, 203, 210, 219 ], [ 8, 15, 36, 43, 64, 71, 92, 99, 120, 127, 148, 155, 176, 183, 204, 211, 220 ], [ 9, 16, 37, 44, 65, 72, 93, 100, 121, 128, 149, 156, 177, 184, 205, 212, 219, 220 ], [ 10, 17, 38, 45, 66, 73, 94, 101, 122, 129, 150, 157, 178, 185, 206, 213 ], [ 11, 18, 39, 46, 67, 74, 95, 102, 123, 130, 151, 158, 179, 186, 207, 214, 219 ], [ 12, 19, 40, 47, 68, 75, 96, 103, 124, 131, 152, 159, 180, 187, 208, 215, 220 ], [ 13, 20, 41, 48, 69, 76, 97, 104, 125, 132, 153, 160, 181, 188, 209, 216, 219, 220 ], [ 21, 28, 35, 42, 77, 84, 91, 98, 133, 140, 147, 154, 189, 196, 203, 210, 221 ], [ 22, 29, 36, 43, 78, 85, 92, 99, 134, 141, 148, 155, 190, 197, 204, 211 ], [ 23, 30, 37, 44, 79, 86, 93, 100, 135, 142, 149, 156, 191, 198, 205, 212, 221 ], [ 24, 31, 38, 45, 80, 87, 94, 101, 136, 143, 150, 157, 192, 199, 206, 213 ], [ 25, 32, 39, 46, 81, 88, 95, 102, 137, 144, 151, 158, 193, 200, 207, 214, 221 ], [ 26, 33, 40, 47, 82, 89, 96, 103, 138, 145, 152, 159, 194, 201, 208, 215 ], [ 27, 34, 41, 48, 83, 90, 97, 104, 139, 146, 153, 160, 195, 202, 209, 216, 221 ], [ 49, 56, 63, 70, 77, 84, 91, 98, 161, 168, 175, 182, 189, 196, 203, 210 ], [ 50, 57, 64, 71, 78, 85, 92, 99, 162, 169, 176, 183, 190, 197, 204, 211, 222 ], [ 51, 58, 65, 72, 79, 86, 93, 100, 163, 170, 177, 184, 191, 198, 205, 212, 222 ], [ 52, 59, 66, 73, 80, 87, 94, 101, 164, 171, 178, 185, 192, 199, 206, 213 ], [ 53, 60, 67, 74, 81, 88, 95, 102, 165, 172, 179, 186, 193, 200, 207, 214 ], [ 54, 61, 68, 75, 82, 89, 96, 103, 166, 173, 180, 187, 194, 201, 208, 215, 222 ], [ 55, 62, 69, 76, 83, 90, 97, 104, 167, 174, 181, 188, 195, 202, 209, 216, 222 ], [ 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196, 203, 210 ], [ 106, 113, 120, 127, 134, 141, 148, 155, 162, 169, 176, 183, 190, 197, 204, 211, 223 ], [ 107, 114, 121, 128, 135, 142, 149, 156, 163, 170, 177, 184, 191, 198, 205, 212, 223 ], [ 108, 115, 122, 129, 136, 143, 150, 157, 164, 171, 178, 185, 192, 199, 206, 213, 224 ], [ 109, 116, 123, 130, 137, 144, 151, 158, 165, 172, 179, 186, 193, 200, 207, 214, 224 ], [ 110, 117, 124, 131, 138, 145, 152, 159, 166, 173, 180, 187, 194, 201, 208, 215, 223, 224 ], [ 111, 118, 125, 132, 139, 146, 153, 160, 167, 174, 181, 188, 195, 202, 209, 216, 223, 224 ] ], "Z": [ [ 0, 2, 4, 6, 217 ], [ 1, 2, 5, 6, 29, 30, 31, 32, 134, 135, 136, 137, 218, 223, 224 ], [ 3, 4, 5, 6, 218 ], [ 8, 9, 12, 13, 220 ], [ 7, 9, 11, 13, 219 ], [ 14, 16, 18, 20, 217, 219 ], [ 15, 16, 19, 20, 29, 30, 31, 32, 134, 135, 136, 137, 218, 220, 223, 224 ], [ 7, 9, 10, 12, 17, 18, 19, 20, 218, 219 ], [ 21, 23, 25, 27, 221 ], [ 22, 23, 26, 27, 134, 135, 136, 137, 143, 144, 145, 146, 218, 223 ], [ 24, 25, 26, 27, 143, 144, 145, 146, 218, 224 ], [ 28, 30, 32, 34, 217, 221 ], [ 31, 32, 33, 34, 143, 144, 145, 146, 224 ], [ 35, 37, 39, 41, 219, 221 ], [ 36, 37, 40, 41, 134, 135, 136, 137, 143, 144, 145, 146, 218, 220, 223 ], [ 7, 9, 10, 12, 38, 39, 40, 41, 143, 144, 145, 146, 218, 219, 224 ], [ 42, 44, 46, 48, 217, 219, 221 ], [ 29, 30, 31, 32, 43, 44, 47, 48, 143, 144, 145, 146, 220, 224 ], [ 7, 9, 10, 12, 45, 46, 47, 48, 143, 144, 145, 146, 219, 224 ], [ 49, 51, 53, 55, 147, 149, 151, 153, 161, 162, 165, 166, 219, 221, 222, 223 ], [ 50, 51, 54, 55, 222 ], [ 52, 53, 54, 55, 143, 144, 145, 146, 199, 200, 201, 202 ], [ 56, 58, 60, 62, 147, 149, 151, 153, 161, 162, 165, 166, 217, 219, 221, 222, 223 ], [ 29, 30, 31, 32, 57, 58, 61, 62, 134, 135, 136, 137, 218, 222, 223, 224 ], [ 59, 60, 61, 62, 143, 144, 145, 146, 199, 200, 201, 202, 218 ], [ 63, 65, 67, 69, 147, 149, 151, 153, 161, 162, 165, 166, 221, 222, 223 ], [ 64, 65, 68, 69, 220, 222 ], [ 7, 9, 10, 12, 66, 67, 68, 69, 143, 144, 145, 146, 199, 200, 201, 202, 219 ], [ 70, 72, 74, 76, 147, 149, 151, 153, 161, 162, 165, 166, 217, 221, 222, 223 ], [ 29, 30, 31, 32, 71, 72, 75, 76, 134, 135, 136, 137, 218, 220, 222, 223, 224 ], [ 7, 9, 10, 12, 73, 74, 75, 76, 143, 144, 145, 146, 199, 200, 201, 202, 218, 219 ], [ 77, 79, 81, 83, 147, 149, 151, 153, 161, 162, 165, 166, 219, 222, 223 ], [ 78, 79, 82, 83, 134, 135, 136, 137, 143, 144, 145, 146, 218, 222, 223 ], [ 80, 81, 82, 83, 199, 200, 201, 202, 218, 224 ], [ 84, 86, 88, 90, 147, 149, 151, 153, 161, 162, 165, 166, 217, 219, 222, 223 ], [ 29, 30, 31, 32, 85, 86, 89, 90, 143, 144, 145, 146, 222, 224 ], [ 87, 88, 89, 90, 199, 200, 201, 202, 224 ], [ 91, 93, 95, 97, 147, 149, 151, 153, 161, 162, 165, 166, 222, 223 ], [ 92, 93, 96, 97, 134, 135, 136, 137, 143, 144, 145, 146, 218, 220, 222, 223 ], [ 7, 9, 10, 12, 94, 95, 96, 97, 199, 200, 201, 202, 218, 219, 224 ], [ 98, 100, 102, 104, 147, 149, 151, 153, 161, 162, 165, 166, 217, 222, 223 ], [ 29, 30, 31, 32, 99, 100, 103, 104, 143, 144, 145, 146, 220, 222, 224 ], [ 7, 9, 10, 12, 101, 102, 103, 104, 199, 200, 201, 202, 219, 224 ], [ 105, 107, 109, 111, 147, 149, 151, 153, 219, 221 ], [ 106, 107, 110, 111, 223 ], [ 108, 109, 110, 111, 224 ], [ 112, 114, 116, 118, 147, 149, 151, 153, 217, 219, 221 ], [ 29, 30, 31, 32, 113, 114, 117, 118, 134, 135, 136, 137, 218, 224 ], [ 115, 116, 117, 118, 218, 224 ], [ 119, 121, 123, 125, 147, 149, 151, 153, 221 ], [ 120, 121, 124, 125, 220, 223 ], [ 7, 9, 10, 12, 122, 123, 124, 125, 219, 224 ], [ 126, 128, 130, 132, 147, 149, 151, 153, 217, 221 ], [ 29, 30, 31, 32, 127, 128, 131, 132, 134, 135, 136, 137, 218, 220, 224 ], [ 7, 9, 10, 12, 129, 130, 131, 132, 218, 219, 224 ], [ 133, 135, 137, 139, 147, 149, 151, 153, 219 ], [ 136, 137, 138, 139, 143, 144, 145, 146, 218 ], [ 140, 142, 144, 146, 147, 149, 151, 153, 217, 219 ], [ 29, 30, 31, 32, 141, 142, 143, 144, 223, 224 ], [ 134, 135, 136, 137, 143, 144, 145, 146, 148, 149, 152, 153, 218, 220 ], [ 7, 9, 10, 12, 143, 144, 145, 146, 150, 151, 152, 153, 218, 219 ], [ 147, 149, 151, 153, 154, 156, 158, 160, 217 ], [ 29, 30, 31, 32, 143, 144, 145, 146, 155, 156, 159, 160, 220, 223, 224 ], [ 7, 9, 10, 12, 143, 144, 145, 146, 157, 158, 159, 160, 219 ], [ 162, 163, 166, 167, 222, 223 ], [ 143, 144, 145, 146, 164, 165, 166, 167, 199, 200, 201, 202, 224 ], [ 161, 162, 165, 166, 168, 170, 172, 174, 217, 222, 223 ], [ 29, 30, 31, 32, 134, 135, 136, 137, 169, 170, 173, 174, 218, 222, 224 ], [ 143, 144, 145, 146, 171, 172, 173, 174, 199, 200, 201, 202, 218, 224 ], [ 161, 162, 165, 166, 175, 177, 179, 181, 219, 222, 223 ], [ 176, 177, 180, 181, 220, 222, 223 ], [ 7, 9, 10, 12, 143, 144, 145, 146, 178, 179, 180, 181, 199, 200, 201, 202, 219, 224 ], [ 161, 162, 165, 166, 182, 184, 186, 188, 217, 219, 222, 223 ], [ 29, 30, 31, 32, 134, 135, 136, 137, 183, 184, 187, 188, 218, 220, 222, 224 ], [ 7, 9, 10, 12, 143, 144, 145, 146, 185, 186, 187, 188, 199, 200, 201, 202, 218, 219, 224 ], [ 161, 162, 165, 166, 189, 191, 193, 195, 221, 222, 223 ], [ 134, 135, 136, 137, 143, 144, 145, 146, 190, 191, 194, 195, 218, 222 ], [ 192, 193, 194, 195, 199, 200, 201, 202, 218 ], [ 161, 162, 165, 166, 196, 198, 200, 202, 217, 221, 222, 223 ], [ 29, 30, 31, 32, 143, 144, 145, 146, 197, 198, 201, 202, 222, 223, 224 ], [ 161, 162, 165, 166, 203, 205, 207, 209, 219, 221, 222, 223 ], [ 134, 135, 136, 137, 143, 144, 145, 146, 204, 205, 208, 209, 218, 220, 222 ], [ 7, 9, 10, 12, 199, 200, 201, 202, 206, 207, 208, 209, 218, 219 ], [ 161, 162, 165, 166, 210, 212, 214, 216, 217, 219, 221, 222, 223 ], [ 29, 30, 31, 32, 143, 144, 145, 146, 211, 212, 215, 216, 220, 222, 223, 224 ], [ 7, 9, 10, 12, 199, 200, 201, 202, 213, 214, 215, 216, 219 ] ] }, "distance": { "d": 3, "X": { "value": 3, "confidence": "upper_bound", "witness": [ 113, 116, 118 ], "witness_provenance": { "found_by": [ "@dorakingx" ], "date": "2026-09-15", "found_at_samples": 20000, "survived_samples": 20000000, "tool": "gf2_fast RIS ladder (verify/gf2_fast.cpp), both sides searched jointly", "seeds": [ 777001, 777138, 777275, 777412, 777549 ] } }, "Z": { "value": 3, "confidence": "upper_bound", "witness": [ 0, 203, 210 ], "witness_provenance": { "found_by": [ "@dorakingx" ], "date": "2026-09-15", "found_at_samples": 4000, "survived_samples": 20000000, "tool": "gf2_fast RIS in the packaging search; the ladder searched both sides jointly", "seeds": [ 7, 777001, 777138, 777275, 777412, 777549 ] } } }, "provenance": { "authors": [ "@dorakingx" ], "construction": "Reduction of the board's codes/232-104-3.json ([[232,104,3]]) by 7 qubits at unchanged k, unchanged check-weight class and unchanged locality class, by one move applied to a fixpoint: graft a qubit away using a low-weight element of the stabilizer ROW SPACE. Let S be an element of the row space of H_X with support T and let a be in T. Replace one generator of the subset summing to S by S itself -- the span is unchanged, that generator being S plus the rest of the subset -- and add S into every other X row meeting a, so that a survives only in S. Then apply the CNOT fan-out from a to T\\{a}, which acts on the check matrices as H_X[:, b] ^= H_X[:, a] and H_Z[:, a] ^= H_Z[:, b] for each b in T\\{a}. Only S still meets a, so the first substitution turns S into the weight-1 stabilizer X_a and leaves every other X row alone; the second zeroes column a of H_Z, because commutation forces every Z row to meet T an even number of times. Qubit a is then disentangled and is deleted: n -> n-1, k unchanged, and the Z rows only LOSE an index, so their weights and support diameters cannot rise. The weight of S separates three regimes, and the difference is entirely in the clearing step R ^= S. |S| = 1: nothing is added anywhere, so the weights, the radius AND THE DISTANCE are all preserved exactly, and no distance search is needed -- this is the row-space form of the weight-1 stabilizer cleanup of Liang, Eberhardt and Chen (arXiv:2504.08887 Sec. III D step 4), which the implementation in research/local2d/boundary_engine.py applies only to literal weight-1 generator ROWS. |S| = 2: R loses a and toggles one other index, so its weight changes by 0 or -2 and can never rise. |S| >= 3: R can grow, so every touched row is checked against the code's weight class and its locality radius, measured in the source's own layout. This generalises the capped merge-graft I introduced with [[454,8,17]], which only ever built S from the generators a single qubit happens to lie in. Here the accepted grafts were |S| = 6: 1, |S| = 7: 2, |S| = 8: 4. Low-weight row-space elements are enumerated as sums of at most three checks with a connected overlap pattern. Every graft with |S| >= 2 was accepted only if k was unchanged and a bit-packed RIS search found nothing lighter than 3, screened once and confirmed twice with independent seeds; those in-loop rungs are a filter, not the evidence. QUBIT POSITIONS ARE THE SOURCE'S: every surviving qubit keeps the coordinate it has in codes/232-104-3.json, so the layout is inherited rather than re-derived, and the reduction only deletes. The surviving qubits' indices into the source numbering are [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]... (full list implied by the coordinates in this file).", "references": [ "arXiv:2504.08887" ], "date": "2026-09-15", "notes": "Derived from codes/232-104-3.json, not an independent construction; the source's authors are @mathysrennela and the reduction is mine. The gate reports no exact duplicate and no WL-equivalent entry. It dominates [[232,104,3]] on (n, k, d, w), which therefore leaves the frontier of every cell the two share. Distance is a witness-backed upper bound: the deepest null result is 20000000 fresh-seed RIS trials. Literature novelty is unverified.", "model": "Claude Opus 5 (Claude Code)", "origin": "submission", "novelty": "unknown" }, "family": "hypergraph-product" }