{ "schema_version": "0.4", "name": "[[270,10,24]] weight-8 coprime bivariate-bicycle code", "code_type": "CSS", "n": 270, "k": 10, "checks": { "X": [ [ 0, 8, 27, 78, 135, 148, 185, 219 ], [ 1, 9, 28, 79, 136, 149, 186, 215 ], [ 2, 5, 29, 75, 137, 145, 187, 216 ], [ 3, 6, 25, 76, 138, 146, 188, 217 ], [ 4, 7, 26, 77, 139, 147, 189, 218 ], [ 5, 13, 32, 83, 140, 153, 190, 224 ], [ 6, 14, 33, 84, 141, 154, 191, 220 ], [ 7, 10, 34, 80, 142, 150, 192, 221 ], [ 8, 11, 30, 81, 143, 151, 193, 222 ], [ 9, 12, 31, 82, 144, 152, 194, 223 ], [ 10, 18, 37, 88, 145, 158, 195, 229 ], [ 11, 19, 38, 89, 146, 159, 196, 225 ], [ 12, 15, 39, 85, 147, 155, 197, 226 ], [ 13, 16, 35, 86, 148, 156, 198, 227 ], [ 14, 17, 36, 87, 149, 157, 199, 228 ], [ 15, 23, 42, 93, 150, 163, 200, 234 ], [ 16, 24, 43, 94, 151, 164, 201, 230 ], [ 17, 20, 44, 90, 152, 160, 202, 231 ], [ 18, 21, 40, 91, 153, 161, 203, 232 ], [ 19, 22, 41, 92, 154, 162, 204, 233 ], [ 20, 28, 47, 98, 155, 168, 205, 239 ], [ 21, 29, 48, 99, 156, 169, 206, 235 ], [ 22, 25, 49, 95, 157, 165, 207, 236 ], [ 23, 26, 45, 96, 158, 166, 208, 237 ], [ 24, 27, 46, 97, 159, 167, 209, 238 ], [ 25, 33, 52, 103, 160, 173, 210, 244 ], [ 26, 34, 53, 104, 161, 174, 211, 240 ], [ 27, 30, 54, 100, 162, 170, 212, 241 ], [ 28, 31, 50, 101, 163, 171, 213, 242 ], [ 29, 32, 51, 102, 164, 172, 214, 243 ], [ 30, 38, 57, 108, 165, 178, 215, 249 ], [ 31, 39, 58, 109, 166, 179, 216, 245 ], [ 32, 35, 59, 105, 167, 175, 217, 246 ], [ 33, 36, 55, 106, 168, 176, 218, 247 ], [ 34, 37, 56, 107, 169, 177, 219, 248 ], [ 35, 43, 62, 113, 170, 183, 220, 254 ], [ 36, 44, 63, 114, 171, 184, 221, 250 ], [ 37, 40, 64, 110, 172, 180, 222, 251 ], [ 38, 41, 60, 111, 173, 181, 223, 252 ], [ 39, 42, 61, 112, 174, 182, 224, 253 ], [ 40, 48, 67, 118, 175, 188, 225, 259 ], [ 41, 49, 68, 119, 176, 189, 226, 255 ], [ 42, 45, 69, 115, 177, 185, 227, 256 ], [ 43, 46, 65, 116, 178, 186, 228, 257 ], [ 44, 47, 66, 117, 179, 187, 229, 258 ], [ 45, 53, 72, 123, 180, 193, 230, 264 ], [ 46, 54, 73, 124, 181, 194, 231, 260 ], [ 47, 50, 74, 120, 182, 190, 232, 261 ], [ 48, 51, 70, 121, 183, 191, 233, 262 ], [ 49, 52, 71, 122, 184, 192, 234, 263 ], [ 50, 58, 77, 128, 185, 198, 235, 269 ], [ 51, 59, 78, 129, 186, 199, 236, 265 ], [ 52, 55, 79, 125, 187, 195, 237, 266 ], [ 53, 56, 75, 126, 188, 196, 238, 267 ], [ 54, 57, 76, 127, 189, 197, 239, 268 ], [ 55, 63, 82, 133, 139, 190, 203, 240 ], [ 56, 64, 83, 134, 135, 191, 204, 241 ], [ 57, 60, 84, 130, 136, 192, 200, 242 ], [ 58, 61, 80, 131, 137, 193, 201, 243 ], [ 59, 62, 81, 132, 138, 194, 202, 244 ], [ 3, 60, 68, 87, 144, 195, 208, 245 ], [ 4, 61, 69, 88, 140, 196, 209, 246 ], [ 0, 62, 65, 89, 141, 197, 205, 247 ], [ 1, 63, 66, 85, 142, 198, 206, 248 ], [ 2, 64, 67, 86, 143, 199, 207, 249 ], [ 8, 65, 73, 92, 149, 200, 213, 250 ], [ 9, 66, 74, 93, 145, 201, 214, 251 ], [ 5, 67, 70, 94, 146, 202, 210, 252 ], [ 6, 68, 71, 90, 147, 203, 211, 253 ], [ 7, 69, 72, 91, 148, 204, 212, 254 ], [ 13, 70, 78, 97, 154, 205, 218, 255 ], [ 14, 71, 79, 98, 150, 206, 219, 256 ], [ 10, 72, 75, 99, 151, 207, 215, 257 ], [ 11, 73, 76, 95, 152, 208, 216, 258 ], [ 12, 74, 77, 96, 153, 209, 217, 259 ], [ 18, 75, 83, 102, 159, 210, 223, 260 ], [ 19, 76, 84, 103, 155, 211, 224, 261 ], [ 15, 77, 80, 104, 156, 212, 220, 262 ], [ 16, 78, 81, 100, 157, 213, 221, 263 ], [ 17, 79, 82, 101, 158, 214, 222, 264 ], [ 23, 80, 88, 107, 164, 215, 228, 265 ], [ 24, 81, 89, 108, 160, 216, 229, 266 ], [ 20, 82, 85, 109, 161, 217, 225, 267 ], [ 21, 83, 86, 105, 162, 218, 226, 268 ], [ 22, 84, 87, 106, 163, 219, 227, 269 ], [ 28, 85, 93, 112, 135, 169, 220, 233 ], [ 29, 86, 94, 113, 136, 165, 221, 234 ], [ 25, 87, 90, 114, 137, 166, 222, 230 ], [ 26, 88, 91, 110, 138, 167, 223, 231 ], [ 27, 89, 92, 111, 139, 168, 224, 232 ], [ 33, 90, 98, 117, 140, 174, 225, 238 ], [ 34, 91, 99, 118, 141, 170, 226, 239 ], [ 30, 92, 95, 119, 142, 171, 227, 235 ], [ 31, 93, 96, 115, 143, 172, 228, 236 ], [ 32, 94, 97, 116, 144, 173, 229, 237 ], [ 38, 95, 103, 122, 145, 179, 230, 243 ], [ 39, 96, 104, 123, 146, 175, 231, 244 ], [ 35, 97, 100, 124, 147, 176, 232, 240 ], [ 36, 98, 101, 120, 148, 177, 233, 241 ], [ 37, 99, 102, 121, 149, 178, 234, 242 ], [ 43, 100, 108, 127, 150, 184, 235, 248 ], [ 44, 101, 109, 128, 151, 180, 236, 249 ], [ 40, 102, 105, 129, 152, 181, 237, 245 ], [ 41, 103, 106, 125, 153, 182, 238, 246 ], [ 42, 104, 107, 126, 154, 183, 239, 247 ], [ 48, 105, 113, 132, 155, 189, 240, 253 ], [ 49, 106, 114, 133, 156, 185, 241, 254 ], [ 45, 107, 110, 134, 157, 186, 242, 250 ], [ 46, 108, 111, 130, 158, 187, 243, 251 ], [ 47, 109, 112, 131, 159, 188, 244, 252 ], [ 2, 53, 110, 118, 160, 194, 245, 258 ], [ 3, 54, 111, 119, 161, 190, 246, 259 ], [ 4, 50, 112, 115, 162, 191, 247, 255 ], [ 0, 51, 113, 116, 163, 192, 248, 256 ], [ 1, 52, 114, 117, 164, 193, 249, 257 ], [ 7, 58, 115, 123, 165, 199, 250, 263 ], [ 8, 59, 116, 124, 166, 195, 251, 264 ], [ 9, 55, 117, 120, 167, 196, 252, 260 ], [ 5, 56, 118, 121, 168, 197, 253, 261 ], [ 6, 57, 119, 122, 169, 198, 254, 262 ], [ 12, 63, 120, 128, 170, 204, 255, 268 ], [ 13, 64, 121, 129, 171, 200, 256, 269 ], [ 14, 60, 122, 125, 172, 201, 257, 265 ], [ 10, 61, 123, 126, 173, 202, 258, 266 ], [ 11, 62, 124, 127, 174, 203, 259, 267 ], [ 17, 68, 125, 133, 138, 175, 209, 260 ], [ 18, 69, 126, 134, 139, 176, 205, 261 ], [ 19, 65, 127, 130, 135, 177, 206, 262 ], [ 15, 66, 128, 131, 136, 178, 207, 263 ], [ 16, 67, 129, 132, 137, 179, 208, 264 ], [ 3, 22, 73, 130, 143, 180, 214, 265 ], [ 4, 23, 74, 131, 144, 181, 210, 266 ], [ 0, 24, 70, 132, 140, 182, 211, 267 ], [ 1, 20, 71, 133, 141, 183, 212, 268 ], [ 2, 21, 72, 134, 142, 184, 213, 269 ] ], "Z": [ [ 0, 56, 85, 127, 135, 197, 248, 267 ], [ 1, 57, 86, 128, 136, 198, 249, 268 ], [ 2, 58, 87, 129, 137, 199, 245, 269 ], [ 3, 59, 88, 125, 138, 195, 246, 265 ], [ 4, 55, 89, 126, 139, 196, 247, 266 ], [ 5, 61, 90, 132, 137, 140, 202, 253 ], [ 6, 62, 91, 133, 138, 141, 203, 254 ], [ 7, 63, 92, 134, 139, 142, 204, 250 ], [ 8, 64, 93, 130, 135, 143, 200, 251 ], [ 9, 60, 94, 131, 136, 144, 201, 252 ], [ 2, 10, 66, 95, 142, 145, 207, 258 ], [ 3, 11, 67, 96, 143, 146, 208, 259 ], [ 4, 12, 68, 97, 144, 147, 209, 255 ], [ 0, 13, 69, 98, 140, 148, 205, 256 ], [ 1, 14, 65, 99, 141, 149, 206, 257 ], [ 7, 15, 71, 100, 147, 150, 212, 263 ], [ 8, 16, 72, 101, 148, 151, 213, 264 ], [ 9, 17, 73, 102, 149, 152, 214, 260 ], [ 5, 18, 74, 103, 145, 153, 210, 261 ], [ 6, 19, 70, 104, 146, 154, 211, 262 ], [ 12, 20, 76, 105, 152, 155, 217, 268 ], [ 13, 21, 77, 106, 153, 156, 218, 269 ], [ 14, 22, 78, 107, 154, 157, 219, 265 ], [ 10, 23, 79, 108, 150, 158, 215, 266 ], [ 11, 24, 75, 109, 151, 159, 216, 267 ], [ 17, 25, 81, 110, 138, 157, 160, 222 ], [ 18, 26, 82, 111, 139, 158, 161, 223 ], [ 19, 27, 83, 112, 135, 159, 162, 224 ], [ 15, 28, 84, 113, 136, 155, 163, 220 ], [ 16, 29, 80, 114, 137, 156, 164, 221 ], [ 22, 30, 86, 115, 143, 162, 165, 227 ], [ 23, 31, 87, 116, 144, 163, 166, 228 ], [ 24, 32, 88, 117, 140, 164, 167, 229 ], [ 20, 33, 89, 118, 141, 160, 168, 225 ], [ 21, 34, 85, 119, 142, 161, 169, 226 ], [ 27, 35, 91, 120, 148, 167, 170, 232 ], [ 28, 36, 92, 121, 149, 168, 171, 233 ], [ 29, 37, 93, 122, 145, 169, 172, 234 ], [ 25, 38, 94, 123, 146, 165, 173, 230 ], [ 26, 39, 90, 124, 147, 166, 174, 231 ], [ 32, 40, 96, 125, 153, 172, 175, 237 ], [ 33, 41, 97, 126, 154, 173, 176, 238 ], [ 34, 42, 98, 127, 150, 174, 177, 239 ], [ 30, 43, 99, 128, 151, 170, 178, 235 ], [ 31, 44, 95, 129, 152, 171, 179, 236 ], [ 37, 45, 101, 130, 158, 177, 180, 242 ], [ 38, 46, 102, 131, 159, 178, 181, 243 ], [ 39, 47, 103, 132, 155, 179, 182, 244 ], [ 35, 48, 104, 133, 156, 175, 183, 240 ], [ 36, 49, 100, 134, 157, 176, 184, 241 ], [ 0, 42, 50, 106, 163, 182, 185, 247 ], [ 1, 43, 51, 107, 164, 183, 186, 248 ], [ 2, 44, 52, 108, 160, 184, 187, 249 ], [ 3, 40, 53, 109, 161, 180, 188, 245 ], [ 4, 41, 54, 105, 162, 181, 189, 246 ], [ 5, 47, 55, 111, 168, 187, 190, 252 ], [ 6, 48, 56, 112, 169, 188, 191, 253 ], [ 7, 49, 57, 113, 165, 189, 192, 254 ], [ 8, 45, 58, 114, 166, 185, 193, 250 ], [ 9, 46, 59, 110, 167, 186, 194, 251 ], [ 10, 52, 60, 116, 173, 192, 195, 257 ], [ 11, 53, 61, 117, 174, 193, 196, 258 ], [ 12, 54, 62, 118, 170, 194, 197, 259 ], [ 13, 50, 63, 119, 171, 190, 198, 255 ], [ 14, 51, 64, 115, 172, 191, 199, 256 ], [ 15, 57, 65, 121, 178, 197, 200, 262 ], [ 16, 58, 66, 122, 179, 198, 201, 263 ], [ 17, 59, 67, 123, 175, 199, 202, 264 ], [ 18, 55, 68, 124, 176, 195, 203, 260 ], [ 19, 56, 69, 120, 177, 196, 204, 261 ], [ 20, 62, 70, 126, 183, 202, 205, 267 ], [ 21, 63, 71, 127, 184, 203, 206, 268 ], [ 22, 64, 72, 128, 180, 204, 207, 269 ], [ 23, 60, 73, 129, 181, 200, 208, 265 ], [ 24, 61, 74, 125, 182, 201, 209, 266 ], [ 25, 67, 75, 131, 137, 188, 207, 210 ], [ 26, 68, 76, 132, 138, 189, 208, 211 ], [ 27, 69, 77, 133, 139, 185, 209, 212 ], [ 28, 65, 78, 134, 135, 186, 205, 213 ], [ 29, 66, 79, 130, 136, 187, 206, 214 ], [ 1, 30, 72, 80, 142, 193, 212, 215 ], [ 2, 31, 73, 81, 143, 194, 213, 216 ], [ 3, 32, 74, 82, 144, 190, 214, 217 ], [ 4, 33, 70, 83, 140, 191, 210, 218 ], [ 0, 34, 71, 84, 141, 192, 211, 219 ], [ 6, 35, 77, 85, 147, 198, 217, 220 ], [ 7, 36, 78, 86, 148, 199, 218, 221 ], [ 8, 37, 79, 87, 149, 195, 219, 222 ], [ 9, 38, 75, 88, 145, 196, 215, 223 ], [ 5, 39, 76, 89, 146, 197, 216, 224 ], [ 11, 40, 82, 90, 152, 203, 222, 225 ], [ 12, 41, 83, 91, 153, 204, 223, 226 ], [ 13, 42, 84, 92, 154, 200, 224, 227 ], [ 14, 43, 80, 93, 150, 201, 220, 228 ], [ 10, 44, 81, 94, 151, 202, 221, 229 ], [ 16, 45, 87, 95, 157, 208, 227, 230 ], [ 17, 46, 88, 96, 158, 209, 228, 231 ], [ 18, 47, 89, 97, 159, 205, 229, 232 ], [ 19, 48, 85, 98, 155, 206, 225, 233 ], [ 15, 49, 86, 99, 156, 207, 226, 234 ], [ 21, 50, 92, 100, 162, 213, 232, 235 ], [ 22, 51, 93, 101, 163, 214, 233, 236 ], [ 23, 52, 94, 102, 164, 210, 234, 237 ], [ 24, 53, 90, 103, 160, 211, 230, 238 ], [ 20, 54, 91, 104, 161, 212, 231, 239 ], [ 26, 55, 97, 105, 167, 218, 237, 240 ], [ 27, 56, 98, 106, 168, 219, 238, 241 ], [ 28, 57, 99, 107, 169, 215, 239, 242 ], [ 29, 58, 95, 108, 165, 216, 235, 243 ], [ 25, 59, 96, 109, 166, 217, 236, 244 ], [ 31, 60, 102, 110, 172, 223, 242, 245 ], [ 32, 61, 103, 111, 173, 224, 243, 246 ], [ 33, 62, 104, 112, 174, 220, 244, 247 ], [ 34, 63, 100, 113, 170, 221, 240, 248 ], [ 30, 64, 101, 114, 171, 222, 241, 249 ], [ 36, 65, 107, 115, 177, 228, 247, 250 ], [ 37, 66, 108, 116, 178, 229, 248, 251 ], [ 38, 67, 109, 117, 179, 225, 249, 252 ], [ 39, 68, 105, 118, 175, 226, 245, 253 ], [ 35, 69, 106, 119, 176, 227, 246, 254 ], [ 41, 70, 112, 120, 182, 233, 252, 255 ], [ 42, 71, 113, 121, 183, 234, 253, 256 ], [ 43, 72, 114, 122, 184, 230, 254, 257 ], [ 44, 73, 110, 123, 180, 231, 250, 258 ], [ 40, 74, 111, 124, 181, 232, 251, 259 ], [ 46, 75, 117, 125, 187, 238, 257, 260 ], [ 47, 76, 118, 126, 188, 239, 258, 261 ], [ 48, 77, 119, 127, 189, 235, 259, 262 ], [ 49, 78, 115, 128, 185, 236, 255, 263 ], [ 45, 79, 116, 129, 186, 237, 256, 264 ], [ 51, 80, 122, 130, 192, 243, 262, 265 ], [ 52, 81, 123, 131, 193, 244, 263, 266 ], [ 53, 82, 124, 132, 194, 240, 264, 267 ], [ 54, 83, 120, 133, 190, 241, 260, 268 ], [ 50, 84, 121, 134, 191, 242, 261, 269 ] ] }, "distance": { "d": 24, "X": { "value": 24, "confidence": "upper_bound", "witness": [ 1, 19, 22, 27, 28, 30, 35, 42, 50, 65, 69, 92, 98, 104, 107, 121, 128, 166, 171, 220, 228, 231, 236, 250 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-27", "found_at_samples": 300000000, "tool": "verify/ris_gpu.py (build/ris_gpu, NVIDIA A40)", "seeds": [ 84128219 ], "survived_samples": 300000000 } }, "Z": { "value": 24, "confidence": "upper_bound", "witness": [ 4, 10, 26, 37, 48, 50, 56, 69, 74, 93, 96, 99, 104, 110, 113, 118, 126, 135, 185, 192, 234, 242, 254, 262 ], "witness_provenance": { "found_by": [ "@vprusso" ], "date": "2026-09-27", "found_at_samples": 300000000, "tool": "verify/ris_gpu.py (build/ris_gpu, NVIDIA A40)", "seeds": [ 84128219 ], "survived_samples": 300000000 } } }, "provenance": { "authors": [ "@vprusso" ], "model": "Claude Fable 5.1 (Claude Code)", "construction": "Periodic bivariate-bicycle code on Z_27 x Z_5 (n = 2*l*m = 270): x = S_27 tensor I_5, y = I_27 tensor S_5 (cyclic shifts), qubit index i*m + j in each block; A = x^0y^0 + x^1y^3 + x^5y^2 + x^15y^3, B = x^0y^0 + x^2y^3 + x^10y^0 + x^16y^4; H_X = [A|B], H_Z = [B^T|A^T] (research/kit/bb.py build_bb(27, 5, [[0, 0], [1, 3], [5, 2], [15, 3]], [[0, 0], [2, 3], [10, 0], [16, 4]])). gcd(27, 5) = 1, so Z_27 x Z_5 is cyclic of order 135 and the code is the cyclic generalized-bicycle code over Z_135 with a(z) = z^0 + z^28 + z^32 + z^123, b(z) = z^0 + z^10 + z^83 + z^124 (CRT relabeling x^a y^b -> z^t, t = a mod 27, t = b mod 5).", "date": "2026-09-27" } }