{ "schema_version": "0.4", "name": "[[329,190,6]] symplectic-halved CPM pair-partition fold", "code_type": "stabilizer", "n": 329, "k": 190, "checks": { "S": [ { "X": [ 17, 58, 102, 177, 218, 263, 314 ], "Z": [ 12, 78, 97, 173, 217, 245, 311 ] }, { "X": [ 18, 59, 103, 178, 219, 264, 315 ], "Z": [ 11, 77, 96, 172, 216, 244, 310 ] }, { "X": [ 19, 60, 104, 179, 220, 265, 316 ], "Z": [ 10, 76, 95, 171, 215, 243, 309 ] }, { "X": [ 20, 61, 105, 180, 221, 266, 317 ], "Z": [ 9, 75, 94, 170, 214, 242, 308 ] }, { "X": [ 21, 62, 106, 181, 222, 267, 318 ], "Z": [ 8, 74, 140, 169, 213, 241, 307 ] }, { "X": [ 22, 63, 107, 182, 223, 268, 319 ], "Z": [ 7, 73, 139, 168, 212, 240, 306 ] }, { "X": [ 23, 64, 108, 183, 224, 269, 320 ], "Z": [ 6, 72, 138, 167, 211, 239, 305 ] }, { "X": [ 24, 65, 109, 184, 225, 270, 321 ], "Z": [ 5, 71, 137, 166, 210, 238, 304 ] }, { "X": [ 25, 66, 110, 185, 226, 271, 322 ], "Z": [ 4, 70, 136, 165, 209, 237, 303 ] }, { "X": [ 26, 67, 111, 186, 227, 272, 323 ], "Z": [ 3, 69, 135, 164, 208, 236, 302 ] }, { "X": [ 27, 68, 112, 187, 228, 273, 324 ], "Z": [ 2, 68, 134, 163, 207, 235, 301 ] }, { "X": [ 28, 69, 113, 141, 229, 274, 325 ], "Z": [ 1, 67, 133, 162, 206, 281, 300 ] }, { "X": [ 29, 70, 114, 142, 230, 275, 326 ], "Z": [ 0, 66, 132, 161, 205, 280, 299 ] }, { "X": [ 30, 71, 115, 143, 231, 276, 327 ], "Z": [ 46, 65, 131, 160, 204, 279, 298 ] }, { "X": [ 31, 72, 116, 144, 232, 277, 328 ], "Z": [ 45, 64, 130, 159, 203, 278, 297 ] }, { "X": [ 32, 73, 117, 145, 233, 278, 282 ], "Z": [ 44, 63, 129, 158, 202, 277, 296 ] }, { "X": [ 33, 74, 118, 146, 234, 279, 283 ], "Z": [ 43, 62, 128, 157, 201, 276, 295 ] }, { "X": [ 34, 75, 119, 147, 188, 280, 284 ], "Z": [ 42, 61, 127, 156, 200, 275, 294 ] }, { "X": [ 35, 76, 120, 148, 189, 281, 285 ], "Z": [ 41, 60, 126, 155, 199, 274, 293 ] }, { "X": [ 36, 77, 121, 149, 190, 235, 286 ], "Z": [ 40, 59, 125, 154, 198, 273, 292 ] }, { "X": [ 37, 78, 122, 150, 191, 236, 287 ], "Z": [ 39, 58, 124, 153, 197, 272, 291 ] }, { "X": [ 38, 79, 123, 151, 192, 237, 288 ], "Z": [ 38, 57, 123, 152, 196, 271, 290 ] }, { "X": [ 39, 80, 124, 152, 193, 238, 289 ], "Z": [ 37, 56, 122, 151, 195, 270, 289 ] }, { "X": [ 40, 81, 125, 153, 194, 239, 290 ], "Z": [ 36, 55, 121, 150, 194, 269, 288 ] }, { "X": [ 41, 82, 126, 154, 195, 240, 291 ], "Z": [ 35, 54, 120, 149, 193, 268, 287 ] }, { "X": [ 42, 83, 127, 155, 196, 241, 292 ], "Z": [ 34, 53, 119, 148, 192, 267, 286 ] }, { "X": [ 43, 84, 128, 156, 197, 242, 293 ], "Z": [ 33, 52, 118, 147, 191, 266, 285 ] }, { "X": [ 44, 85, 129, 157, 198, 243, 294 ], "Z": [ 32, 51, 117, 146, 190, 265, 284 ] }, { "X": [ 45, 86, 130, 158, 199, 244, 295 ], "Z": [ 31, 50, 116, 145, 189, 264, 283 ] }, { "X": [ 46, 87, 131, 159, 200, 245, 296 ], "Z": [ 30, 49, 115, 144, 188, 263, 282 ] }, { "X": [ 0, 88, 132, 160, 201, 246, 297 ], "Z": [ 29, 48, 114, 143, 234, 262, 328 ] }, { "X": [ 1, 89, 133, 161, 202, 247, 298 ], "Z": [ 28, 47, 113, 142, 233, 261, 327 ] }, { "X": [ 2, 90, 134, 162, 203, 248, 299 ], "Z": [ 27, 93, 112, 141, 232, 260, 326 ] }, { "X": [ 3, 91, 135, 163, 204, 249, 300 ], "Z": [ 26, 92, 111, 187, 231, 259, 325 ] }, { "X": [ 4, 92, 136, 164, 205, 250, 301 ], "Z": [ 25, 91, 110, 186, 230, 258, 324 ] }, { "X": [ 5, 93, 137, 165, 206, 251, 302 ], "Z": [ 24, 90, 109, 185, 229, 257, 323 ] }, { "X": [ 6, 47, 138, 166, 207, 252, 303 ], "Z": [ 23, 89, 108, 184, 228, 256, 322 ] }, { "X": [ 7, 48, 139, 167, 208, 253, 304 ], "Z": [ 22, 88, 107, 183, 227, 255, 321 ] }, { "X": [ 8, 49, 140, 168, 209, 254, 305 ], "Z": [ 21, 87, 106, 182, 226, 254, 320 ] }, { "X": [ 9, 50, 94, 169, 210, 255, 306 ], "Z": [ 20, 86, 105, 181, 225, 253, 319 ] }, { "X": [ 10, 51, 95, 170, 211, 256, 307 ], "Z": [ 19, 85, 104, 180, 224, 252, 318 ] }, { "X": [ 11, 52, 96, 171, 212, 257, 308 ], "Z": [ 18, 84, 103, 179, 223, 251, 317 ] }, { "X": [ 12, 53, 97, 172, 213, 258, 309 ], "Z": [ 17, 83, 102, 178, 222, 250, 316 ] }, { "X": [ 13, 54, 98, 173, 214, 259, 310 ], "Z": [ 16, 82, 101, 177, 221, 249, 315 ] }, { "X": [ 14, 55, 99, 174, 215, 260, 311 ], "Z": [ 15, 81, 100, 176, 220, 248, 314 ] }, { "X": [ 15, 56, 100, 175, 216, 261, 312 ], "Z": [ 14, 80, 99, 175, 219, 247, 313 ] }, { "X": [ 16, 57, 101, 176, 217, 262, 313 ], "Z": [ 13, 79, 98, 174, 218, 246, 312 ] }, { "X": [ 14, 75, 135, 165, 234, 236, 316 ], "Z": [ 25, 56, 97, 180, 189, 246, 285 ] }, { "X": [ 15, 76, 136, 166, 188, 237, 317 ], "Z": [ 24, 55, 96, 179, 188, 245, 284 ] }, { "X": [ 16, 77, 137, 167, 189, 238, 318 ], "Z": [ 23, 54, 95, 178, 234, 244, 283 ] }, { "X": [ 17, 78, 138, 168, 190, 239, 319 ], "Z": [ 22, 53, 94, 177, 233, 243, 282 ] }, { "X": [ 18, 79, 139, 169, 191, 240, 320 ], "Z": [ 21, 52, 140, 176, 232, 242, 328 ] }, { "X": [ 19, 80, 140, 170, 192, 241, 321 ], "Z": [ 20, 51, 139, 175, 231, 241, 327 ] }, { "X": [ 20, 81, 94, 171, 193, 242, 322 ], "Z": [ 19, 50, 138, 174, 230, 240, 326 ] }, { "X": [ 21, 82, 95, 172, 194, 243, 323 ], "Z": [ 18, 49, 137, 173, 229, 239, 325 ] }, { "X": [ 22, 83, 96, 173, 195, 244, 324 ], "Z": [ 17, 48, 136, 172, 228, 238, 324 ] }, { "X": [ 23, 84, 97, 174, 196, 245, 325 ], "Z": [ 16, 47, 135, 171, 227, 237, 323 ] }, { "X": [ 24, 85, 98, 175, 197, 246, 326 ], "Z": [ 15, 93, 134, 170, 226, 236, 322 ] }, { "X": [ 25, 86, 99, 176, 198, 247, 327 ], "Z": [ 14, 92, 133, 169, 225, 235, 321 ] }, { "X": [ 26, 87, 100, 177, 199, 248, 328 ], "Z": [ 13, 91, 132, 168, 224, 281, 320 ] }, { "X": [ 27, 88, 101, 178, 200, 249, 282 ], "Z": [ 12, 90, 131, 167, 223, 280, 319 ] }, { "X": [ 28, 89, 102, 179, 201, 250, 283 ], "Z": [ 11, 89, 130, 166, 222, 279, 318 ] }, { "X": [ 29, 90, 103, 180, 202, 251, 284 ], "Z": [ 10, 88, 129, 165, 221, 278, 317 ] }, { "X": [ 30, 91, 104, 181, 203, 252, 285 ], "Z": [ 9, 87, 128, 164, 220, 277, 316 ] }, { "X": [ 31, 92, 105, 182, 204, 253, 286 ], "Z": [ 8, 86, 127, 163, 219, 276, 315 ] }, { "X": [ 32, 93, 106, 183, 205, 254, 287 ], "Z": [ 7, 85, 126, 162, 218, 275, 314 ] }, { "X": [ 33, 47, 107, 184, 206, 255, 288 ], "Z": [ 6, 84, 125, 161, 217, 274, 313 ] }, { "X": [ 34, 48, 108, 185, 207, 256, 289 ], "Z": [ 5, 83, 124, 160, 216, 273, 312 ] }, { "X": [ 35, 49, 109, 186, 208, 257, 290 ], "Z": [ 4, 82, 123, 159, 215, 272, 311 ] }, { "X": [ 36, 50, 110, 187, 209, 258, 291 ], "Z": [ 3, 81, 122, 158, 214, 271, 310 ] }, { "X": [ 37, 51, 111, 141, 210, 259, 292 ], "Z": [ 2, 80, 121, 157, 213, 270, 309 ] }, { "X": [ 38, 52, 112, 142, 211, 260, 293 ], "Z": [ 1, 79, 120, 156, 212, 269, 308 ] }, { "X": [ 39, 53, 113, 143, 212, 261, 294 ], "Z": [ 0, 78, 119, 155, 211, 268, 307 ] }, { "X": [ 40, 54, 114, 144, 213, 262, 295 ], "Z": [ 46, 77, 118, 154, 210, 267, 306 ] }, { "X": [ 41, 55, 115, 145, 214, 263, 296 ], "Z": [ 45, 76, 117, 153, 209, 266, 305 ] }, { "X": [ 42, 56, 116, 146, 215, 264, 297 ], "Z": [ 44, 75, 116, 152, 208, 265, 304 ] }, { "X": [ 43, 57, 117, 147, 216, 265, 298 ], "Z": [ 43, 74, 115, 151, 207, 264, 303 ] }, { "X": [ 44, 58, 118, 148, 217, 266, 299 ], "Z": [ 42, 73, 114, 150, 206, 263, 302 ] }, { "X": [ 45, 59, 119, 149, 218, 267, 300 ], "Z": [ 41, 72, 113, 149, 205, 262, 301 ] }, { "X": [ 46, 60, 120, 150, 219, 268, 301 ], "Z": [ 40, 71, 112, 148, 204, 261, 300 ] }, { "X": [ 0, 61, 121, 151, 220, 269, 302 ], "Z": [ 39, 70, 111, 147, 203, 260, 299 ] }, { "X": [ 1, 62, 122, 152, 221, 270, 303 ], "Z": [ 38, 69, 110, 146, 202, 259, 298 ] }, { "X": [ 2, 63, 123, 153, 222, 271, 304 ], "Z": [ 37, 68, 109, 145, 201, 258, 297 ] }, { "X": [ 3, 64, 124, 154, 223, 272, 305 ], "Z": [ 36, 67, 108, 144, 200, 257, 296 ] }, { "X": [ 4, 65, 125, 155, 224, 273, 306 ], "Z": [ 35, 66, 107, 143, 199, 256, 295 ] }, { "X": [ 5, 66, 126, 156, 225, 274, 307 ], "Z": [ 34, 65, 106, 142, 198, 255, 294 ] }, { "X": [ 6, 67, 127, 157, 226, 275, 308 ], "Z": [ 33, 64, 105, 141, 197, 254, 293 ] }, { "X": [ 7, 68, 128, 158, 227, 276, 309 ], "Z": [ 32, 63, 104, 187, 196, 253, 292 ] }, { "X": [ 8, 69, 129, 159, 228, 277, 310 ], "Z": [ 31, 62, 103, 186, 195, 252, 291 ] }, { "X": [ 9, 70, 130, 160, 229, 278, 311 ], "Z": [ 30, 61, 102, 185, 194, 251, 290 ] }, { "X": [ 10, 71, 131, 161, 230, 279, 312 ], "Z": [ 29, 60, 101, 184, 193, 250, 289 ] }, { "X": [ 11, 72, 132, 162, 231, 280, 313 ], "Z": [ 28, 59, 100, 183, 192, 249, 288 ] }, { "X": [ 12, 73, 133, 163, 232, 281, 314 ], "Z": [ 27, 58, 99, 182, 191, 248, 287 ] }, { "X": [ 13, 74, 134, 164, 233, 235, 315 ], "Z": [ 26, 57, 98, 181, 190, 247, 286 ] }, { "X": [ 19, 68, 98, 174, 221, 278, 297 ], "Z": [ 13, 70, 112, 142, 229, 274, 307 ] }, { "X": [ 20, 69, 99, 175, 222, 279, 298 ], "Z": [ 12, 69, 111, 141, 228, 273, 306 ] }, { "X": [ 21, 70, 100, 176, 223, 280, 299 ], "Z": [ 11, 68, 110, 187, 227, 272, 305 ] }, { "X": [ 22, 71, 101, 177, 224, 281, 300 ], "Z": [ 10, 67, 109, 186, 226, 271, 304 ] }, { "X": [ 23, 72, 102, 178, 225, 235, 301 ], "Z": [ 9, 66, 108, 185, 225, 270, 303 ] }, { "X": [ 24, 73, 103, 179, 226, 236, 302 ], "Z": [ 8, 65, 107, 184, 224, 269, 302 ] }, { "X": [ 25, 74, 104, 180, 227, 237, 303 ], "Z": [ 7, 64, 106, 183, 223, 268, 301 ] }, { "X": [ 26, 75, 105, 181, 228, 238, 304 ], "Z": [ 6, 63, 105, 182, 222, 267, 300 ] }, { "X": [ 27, 76, 106, 182, 229, 239, 305 ], "Z": [ 5, 62, 104, 181, 221, 266, 299 ] }, { "X": [ 28, 77, 107, 183, 230, 240, 306 ], "Z": [ 4, 61, 103, 180, 220, 265, 298 ] }, { "X": [ 29, 78, 108, 184, 231, 241, 307 ], "Z": [ 3, 60, 102, 179, 219, 264, 297 ] }, { "X": [ 30, 79, 109, 185, 232, 242, 308 ], "Z": [ 2, 59, 101, 178, 218, 263, 296 ] }, { "X": [ 31, 80, 110, 186, 233, 243, 309 ], "Z": [ 1, 58, 100, 177, 217, 262, 295 ] }, { "X": [ 32, 81, 111, 187, 234, 244, 310 ], "Z": [ 0, 57, 99, 176, 216, 261, 294 ] }, { "X": [ 33, 82, 112, 141, 188, 245, 311 ], "Z": [ 46, 56, 98, 175, 215, 260, 293 ] }, { "X": [ 34, 83, 113, 142, 189, 246, 312 ], "Z": [ 45, 55, 97, 174, 214, 259, 292 ] }, { "X": [ 35, 84, 114, 143, 190, 247, 313 ], "Z": [ 44, 54, 96, 173, 213, 258, 291 ] }, { "X": [ 36, 85, 115, 144, 191, 248, 314 ], "Z": [ 43, 53, 95, 172, 212, 257, 290 ] }, { "X": [ 37, 86, 116, 145, 192, 249, 315 ], "Z": [ 42, 52, 94, 171, 211, 256, 289 ] }, { "X": [ 38, 87, 117, 146, 193, 250, 316 ], "Z": [ 41, 51, 140, 170, 210, 255, 288 ] }, { "X": [ 39, 88, 118, 147, 194, 251, 317 ], "Z": [ 40, 50, 139, 169, 209, 254, 287 ] }, { "X": [ 40, 89, 119, 148, 195, 252, 318 ], "Z": [ 39, 49, 138, 168, 208, 253, 286 ] }, { "X": [ 41, 90, 120, 149, 196, 253, 319 ], "Z": [ 38, 48, 137, 167, 207, 252, 285 ] }, { "X": [ 42, 91, 121, 150, 197, 254, 320 ], "Z": [ 37, 47, 136, 166, 206, 251, 284 ] }, { "X": [ 43, 92, 122, 151, 198, 255, 321 ], "Z": [ 36, 93, 135, 165, 205, 250, 283 ] }, { "X": [ 44, 93, 123, 152, 199, 256, 322 ], "Z": [ 35, 92, 134, 164, 204, 249, 282 ] }, { "X": [ 45, 47, 124, 153, 200, 257, 323 ], "Z": [ 34, 91, 133, 163, 203, 248, 328 ] }, { "X": [ 46, 48, 125, 154, 201, 258, 324 ], "Z": [ 33, 90, 132, 162, 202, 247, 327 ] }, { "X": [ 0, 49, 126, 155, 202, 259, 325 ], "Z": [ 32, 89, 131, 161, 201, 246, 326 ] }, { "X": [ 1, 50, 127, 156, 203, 260, 326 ], "Z": [ 31, 88, 130, 160, 200, 245, 325 ] }, { "X": [ 2, 51, 128, 157, 204, 261, 327 ], "Z": [ 30, 87, 129, 159, 199, 244, 324 ] }, { "X": [ 3, 52, 129, 158, 205, 262, 328 ], "Z": [ 29, 86, 128, 158, 198, 243, 323 ] }, { "X": [ 4, 53, 130, 159, 206, 263, 282 ], "Z": [ 28, 85, 127, 157, 197, 242, 322 ] }, { "X": [ 5, 54, 131, 160, 207, 264, 283 ], "Z": [ 27, 84, 126, 156, 196, 241, 321 ] }, { "X": [ 6, 55, 132, 161, 208, 265, 284 ], "Z": [ 26, 83, 125, 155, 195, 240, 320 ] }, { "X": [ 7, 56, 133, 162, 209, 266, 285 ], "Z": [ 25, 82, 124, 154, 194, 239, 319 ] }, { "X": [ 8, 57, 134, 163, 210, 267, 286 ], "Z": [ 24, 81, 123, 153, 193, 238, 318 ] }, { "X": [ 9, 58, 135, 164, 211, 268, 287 ], "Z": [ 23, 80, 122, 152, 192, 237, 317 ] }, { "X": [ 10, 59, 136, 165, 212, 269, 288 ], "Z": [ 22, 79, 121, 151, 191, 236, 316 ] }, { "X": [ 11, 60, 137, 166, 213, 270, 289 ], "Z": [ 21, 78, 120, 150, 190, 235, 315 ] }, { "X": [ 12, 61, 138, 167, 214, 271, 290 ], "Z": [ 20, 77, 119, 149, 189, 281, 314 ] }, { "X": [ 13, 62, 139, 168, 215, 272, 291 ], "Z": [ 19, 76, 118, 148, 188, 280, 313 ] }, { "X": [ 14, 63, 140, 169, 216, 273, 292 ], "Z": [ 18, 75, 117, 147, 234, 279, 312 ] }, { "X": [ 15, 64, 94, 170, 217, 274, 293 ], "Z": [ 17, 74, 116, 146, 233, 278, 311 ] }, { "X": [ 16, 65, 95, 171, 218, 275, 294 ], "Z": [ 16, 73, 115, 145, 232, 277, 310 ] }, { "X": [ 17, 66, 96, 172, 219, 276, 295 ], "Z": [ 15, 72, 114, 144, 231, 276, 309 ] }, { "X": [ 18, 67, 97, 173, 220, 277, 296 ], "Z": [ 14, 71, 113, 143, 230, 275, 308 ] } ] }, "distance": { "d": 6, "P": { "value": 6, "confidence": "upper_bound", "witness": { "X": [ 76, 116, 215, 311 ], "Z": [ 130, 215, 263, 311 ] } } }, "provenance": { "construction": "Symplectic-halved CPM pair-partition code: Prop. 4 of arXiv:2609.30069. (J,L,P)=(3,14,47); the CSS parent on n_par=L*P=658 qubits has block (i,l) = C(E[i][l]) over Z_47 in H_X and D[j][l] = -E[j][sigma(l)] mod 47 in H_Z, with E = [[30, 36, 39, 11, 17, 19, 15, 12, 31, 29, 32, 10, 3, 29], [33, 19, 6, 23, 1, 46, 13, 25, 9, 3, 39, 11, 3, 1], [28, 26, 43, 14, 14, 4, 32, 13, 23, 25, 1, 39, 18, 41]] and sigma(l) = l=0->7, l=1->8, l=2->12, l=3->10, l=4->13, l=5->11, l=6->9, l=7->0, l=8->1, l=9->6, l=10->3, l=11->5, l=12->2, l=13->4 (a fixed-point-free involution of the 14 block columns). The parent is folded under pi(l,t) = (sigma(l), -t) to this general stabilizer code S = (A | B) on 329 qubits; k = n - rank S.", "notes": "Fold of a halving-constrained draw; the CSS parent is the symplectic double and is submitted as its own entry. The exponent array solves the pair-partition equations E[i][u]-E[i][v] = -(E[j][sigma(u)]-E[j][sigma(v)]) in a null space of dimension 12 and was hill-climbed inside it, so the CSS condition and the halving identity cannot be broken by a move. k = 190 against the parent's 380. Equivalence to an existing entry was checked by the trusted gate (exact-duplicate and WL-equivalent both null). Literature novelty unverified. Distance is a witness-backed upper bound, not an exact certificate.", "authors": [ "@MathysRennela" ], "references": [ "arXiv:2609.30069", "arXiv:2607.14091" ], "date": "2026-10-01", "model": "Space Bunny Alpha 1.0" }, "family": "pair-partition-cpm" }