{ "schema_version": "0.1", "name": "[[272,152,4]] 1x4 binomial lifted product, check weight10", "code_type": "CSS", "n": 272, "k": 152, "checks": { "X": [ [ 1, 2, 65, 68, 129, 134, 193, 200, 257, 258 ], [ 2, 3, 66, 69, 130, 135, 194, 201, 258, 259 ], [ 3, 4, 67, 70, 131, 136, 195, 202, 259, 260 ], [ 4, 5, 68, 71, 132, 137, 196, 203, 260, 261 ], [ 5, 6, 69, 72, 133, 138, 197, 204, 261, 262 ], [ 6, 7, 70, 73, 134, 139, 198, 205, 262, 263 ], [ 7, 8, 71, 74, 135, 140, 199, 206, 263, 264 ], [ 8, 9, 72, 75, 136, 141, 200, 207, 264, 265 ], [ 9, 10, 73, 76, 137, 142, 192, 201, 265, 266 ], [ 10, 11, 74, 77, 138, 143, 193, 202, 266, 267 ], [ 11, 12, 75, 78, 128, 139, 194, 203, 267, 268 ], [ 12, 13, 76, 79, 129, 140, 195, 204, 268, 269 ], [ 13, 14, 64, 77, 130, 141, 196, 205, 269, 270 ], [ 14, 15, 65, 78, 131, 142, 197, 206, 270, 271 ], [ 0, 15, 66, 79, 132, 143, 198, 207, 256, 271 ], [ 16, 17, 80, 83, 144, 149, 208, 215, 256, 259 ], [ 17, 18, 81, 84, 145, 150, 209, 216, 257, 260 ], [ 19, 20, 83, 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234, 243, 250, 259, 266 ], [ 202, 203, 216, 219, 230, 235, 244, 251, 260, 267 ], [ 203, 204, 217, 220, 231, 236, 245, 252, 261, 268 ], [ 204, 205, 218, 221, 232, 237, 246, 253, 262, 269 ], [ 206, 207, 220, 223, 234, 239, 248, 255, 264, 271 ] ] }, "distance": { "d": 4, "X": { "value": 4, "confidence": "upper_bound", "witness": [ 0, 1, 15, 31 ] }, "Z": { "value": 4, "confidence": "upper_bound", "witness": [ 0, 1, 2, 66 ] } }, "provenance": { "authors": [ "@hirobumii" ], "construction": "G=C16 indexed0..15 with multiplication(x,y)=x+y mod16. L_g[t,u]=1 iff u=g+t; R_h[t,u]=1 iff u=t+h. A=[I+L1,I+L3,I+L5,I+L7], B=[I+R1,I+R3,I+R5,I+R7], binary sums. Sixteen VV(k,j) blocks indexed4*k+j precede one CC block, each size16. X(j) places A(k) on VV(k,j) and B(j) on CC; Z(k) places B(j)^T on VV(k,j) and A(k)^T on CC. Omit one original row per block: X indices within blocks[0,2,3,4], Z[13,9,15,14]. Keep all other rows,60 per sector. Their CC edge omissions partition the16 CC vertices and give degree15.", "references": [ "https://arxiv.org/abs/2111.03654", "https://arxiv.org/abs/2607.28795" ], "date": "2026-10-17", "notes": "Displayed presentation: maximum check weight 10, maximum qubit degree 15, 120 checks and total incidence 1200.", "model": "GPT-6 Astra + GPT-6.1 Sol (recorded multi-agent configuration)" }, "family": "lifted-product" }