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Columns over GF(2): the tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5, k=1..p. 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"provenance": "Found independently in two searches, August 2026: Chris Eur and David Renshaw via AlphaEvolve, and June Huh via the study of Mathieu groups with ChatGPT 5.5 Pro assisting. Huh observed that contracting two elements yields the matroid of the Steiner system S(3,4,10), related to duads and synthemes. Submitted from the discovery record.", "verifier_version": "2.0.0", "submitter": "David Renshaw", "created_at": "2026-08-05 15:25:29", "titles": [ "3" ], "current": true }, { "id": 2, "name": "HSW finite-field spike M_3^5", "n": 14, "rank": 5, "num_bases": 891, "alpha": { "numerator": 8, "denominator": 7, "value": 1.1428571428571428 }, "max_pairs": [ { "i": 0, "j": 1, "counts": { "containBoth": 108, "avoidBoth": 324, "onlyI": 81, "onlyJ": 378 } } ], "max_counts": { "containBoth": 108, "avoidBoth": 324, "onlyI": 81, "onlyJ": 378 }, "representation_type": "matrix", "representation": [ [ 1, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0, 1, 2, 0 ], [ 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1 ] ], "field": "3", "cert_field": null, "cert_matrix": null, "bases": [ [ 0, 1, 2, 5, 8 ], [ 0, 1, 3, 5, 8 ], [ 1, 2, 3, 5, 8 ], [ 0, 1, 4, 5, 8 ], [ 1, 2, 4, 5, 8 ], [ 1, 3, 4, 5, 8 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Columns over GF(3): the tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5, k=1..p. 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12, 16, 17 ], [ 5, 7, 12, 16, 17 ], [ 1, 8, 12, 16, 17 ], [ 2, 8, 12, 16, 17 ], [ 3, 8, 12, 16, 17 ], [ 4, 8, 12, 16, 17 ], [ 5, 8, 12, 16, 17 ], [ 1, 9, 12, 16, 17 ], [ 2, 9, 12, 16, 17 ], [ 3, 9, 12, 16, 17 ], [ 4, 9, 12, 16, 17 ], [ 5, 9, 12, 16, 17 ], [ 1, 2, 13, 16, 17 ], [ 1, 3, 13, 16, 17 ], [ 1, 4, 13, 16, 17 ], [ 1, 5, 13, 16, 17 ], [ 1, 6, 13, 16, 17 ], [ 2, 6, 13, 16, 17 ], [ 3, 6, 13, 16, 17 ], [ 4, 6, 13, 16, 17 ], [ 5, 6, 13, 16, 17 ], [ 1, 7, 13, 16, 17 ], [ 2, 7, 13, 16, 17 ], [ 3, 7, 13, 16, 17 ], [ 4, 7, 13, 16, 17 ], [ 5, 7, 13, 16, 17 ], [ 1, 8, 13, 16, 17 ], [ 2, 8, 13, 16, 17 ], [ 3, 8, 13, 16, 17 ], [ 4, 8, 13, 16, 17 ], [ 5, 8, 13, 16, 17 ], [ 1, 9, 13, 16, 17 ], [ 2, 9, 13, 16, 17 ], [ 3, 9, 13, 16, 17 ], [ 4, 9, 13, 16, 17 ], [ 5, 9, 13, 16, 17 ] ], "discovery": { "date": "2026-08", "discoverers": "Construction: Huh–Schröter–Wang; instantiated over GF(4) for this site", "ai_used": true, "ai_model": "Claude Fable 5 (via Claude Code)", "ai_role": "Assembled and verified the GF(4) instantiation of the spike construction.", "ai_chat_url": null }, "citation": "arXiv:1806.02675, Definition 19", "provenance": "The Huh-Schroter-Wang spike construction (Correlation bounds for fields and matroids, arXiv:1806.02675, Definition 19) instantiated over the non-prime field GF(4): tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5 and all k in GF(4) (entries in the Conway basis, a = [0,1]). The paper fixes a prime p, but the Section 5 counting argument applies verbatim over any finite field; the counts (256, 1536, 256, 1344) match its formulas at q=4. Assembled and verified for this site, August 2026, with AI assistance (Claude Fable 5 via Claude Code).", "verifier_version": "2.0.0", "submitter": "David Renshaw", "created_at": "2026-08-05 17:09:19", "titles": [], "current": false }, { "id": 4, "name": "HSW finite-field spike M_5^5", "n": 22, "rank": 5, "num_bases": 9625, "alpha": { "numerator": 8, "denominator": 7, "value": 1.1428571428571428 }, "max_pairs": [ { "i": 0, "j": 1, "counts": { "containBoth": 500, "avoidBoth": 5000, "onlyI": 625, "onlyJ": 3500 } } ], "max_counts": { "containBoth": 500, "avoidBoth": 5000, "onlyI": 625, "onlyJ": 3500 }, "representation_type": "matrix", "representation": [ [ 1, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0, 1, 2, 3, 4, 0 ], [ 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1 ] ], "field": "5", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2018-06", "discoverers": "June Huh, Benjamin Schröter, Botong Wang", "ai_used": false, "ai_model": null, "ai_role": null, "ai_chat_url": null }, "citation": "arXiv:1806.02675, Definition 19 (JEMS 24 (2022), 1335–1351)", "provenance": "The finite-field spike M_5^5 of Huh, Schroter, and Wang (Correlation bounds for fields and matroids, arXiv:1806.02675, Definition 19; discovered June 2018, no AI use reported in that record). Columns over GF(5): the tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5, k=1..p. Submitted from the literature.", "verifier_version": "2.0.0", "submitter": "David Renshaw", "created_at": "2026-08-05 15:39:09", "titles": [], "current": false }, { "id": 9, "name": null, "n": 22, "rank": 9, "num_bases": 176265, "alpha": { "numerator": 4664, "denominator": 4007, "value": 1.1639630646368855 }, "max_pairs": [ { "i": 20, "j": 21, "counts": { "containBoth": 27500, "avoidBoth": 53000, "onlyI": 80140, "onlyJ": 15625 } } ], "max_counts": { "containBoth": 27500, "avoidBoth": 53000, "onlyI": 80140, "onlyJ": 15625 }, "representation_type": "matrix", "representation": [ [ 0, 4, 4, 3, 3, 3, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 3, 1, 4, 1, 4, 0, 4, 0, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 2, 3, 4, 1, 0, 3, 3, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 3, 3, 3, 0, 3, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 1, 4, 1, 4, 0, 4, 0, 3, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 4, 1, 0, 3, 3, 2, 0, 0 ], [ 3, 1, 0, 2, 3, 2, 3, 1, 1, 3, 0, 2, 2, 0, 2, 3, 0, 3, 3, 3, 2, 1 ] ], "field": "5", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08-05", "discoverers": "Chris Eur", "ai_used": true, "ai_model": "ChatGPT 5.6 Sol Ultra", "ai_role": "Found by prompting Codex to look for variations of the double-free extension construction in small finite fields.", "ai_chat_url": null }, "citation": null, "provenance": "This is a (not free) coextension of the free extension of the self-direct-sum of the matroid of the complete graph on 5 vertices.", "verifier_version": "2.0.0", "submitter": "Christopher Eur", "created_at": "2026-08-06 20:33:30", "titles": [], "current": false }, { "id": 15, "name": "Three-block sparse affine-lift GF(5) record", "n": 23, "rank": 13, "num_bases": 234056, "alpha": { "numerator": 15750, "denominator": 13283, "value": 1.185726116088233 }, "max_pairs": [ { "i": 21, "j": 22, "counts": { "containBoth": 58800, "avoidBoth": 47040, "onlyI": 106264, "onlyJ": 21952 } } ], "max_counts": { "containBoth": 58800, "avoidBoth": 47040, "onlyI": 106264, "onlyJ": 21952 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ], [ 0, 0, 1, 0, 3, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ], [ 0, 0, 0, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 4, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 2, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 4, 1, 2, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 3, 3, 0 ], [ 0, 3, 1, 3, 1, 1, 3, 4, 3, 3, 2, 0, 3, 2, 3, 3, 1, 0, 2, 3, 3, 0, 1 ] ], "field": "5", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08-09", "discoverers": "Jeewon Kim", "ai_used": true, "ai_model": "GPT-5.6 Sol Ultra", "ai_role": "AI performed the computational construction search, optimized the finite-field block and affine-lift parameters, and exhaustively verified every basis and every eligible element pair. Two independent implementations reproduced the exact correlation constant.", "ai_chat_url": "https://chatgpt.com/share/6a78818b-8e8c-83ee-a900-83f231ef89ec" }, "citation": null, "provenance": "The construction has the block form\r\n\r\n[C 0 0 u 0\r\n0 C 0 u 0\r\n0 0 C u 0\r\nh1 h2 h3 0 1],\r\n\r\nwhere C is the rank-4, 7-column seed\r\n\r\nC =\r\n[1 0 0 0 1 1 1\r\n0 1 0 0 1 3 0\r\n0 0 1 0 3 4 1\r\n0 0 0 1 3 3 3],\r\n\r\nu = [1,2,2,3]^T,\r\n\r\nh1 = [0,3,1,3,1,1,3],\r\nh2 = [4,3,3,2,0,3,2],\r\nh3 = [3,3,1,0,2,3,3].\r\n\r\nThe final two columns are the distinguished pair. The matroid has rank 13 and 23 elements. Exhaustive basis enumeration and an all-pairs check give the exact global correlation constant\r\n\r\nalpha-bar = 15750/13283,\r\n\r\nattained by the final two columns.", "verifier_version": "2.0.0", "submitter": "Jeewon Kim", "created_at": "2026-08-09 14:04:27", "titles": [], "current": false }, { "id": 16, "name": "Three-block sparse affine-lift GF(7) record", "n": 23, "rank": 13, "num_bases": 77827, "alpha": { "numerator": 40500, "denominator": 33827, "value": 1.1972684541933958 }, "max_pairs": [ { "i": 21, "j": 22, "counts": { "containBoth": 18000, "avoidBoth": 18000, "onlyI": 33827, "onlyJ": 8000 } } ], "max_counts": { "containBoth": 18000, "avoidBoth": 18000, "onlyI": 33827, "onlyJ": 8000 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 2, 1, 0, 0, 5, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 2, 0, 1, 0, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0 ], [ 3, 0, 0, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 5, 1, 6, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 4, 3, 1, 0, 0, 0, 0, 0, 0, 0, 6, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 5, 1, 6, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 4, 3, 1, 6, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 1, 3, 3, 3, 3, 0 ], [ 2, 1, 6, 0, 1, 1, 4, 6, 0, 6, 6, 2, 6, 4, 5, 2, 5, 2, 2, 0, 4, 0, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08-09", "discoverers": "Jeewon Kim", "ai_used": true, "ai_model": "GPT-5.6 Sol Ultra", "ai_role": "AI performed the computational construction search, optimized the finite-field block and affine-lift parameters, and exhaustively verified every basis and every eligible element pair. Two independent implementations reproduced the exact correlation constant.", "ai_chat_url": "https://chatgpt.com/share/6a78818b-8e8c-83ee-a900-83f231ef89ec" }, "citation": null, "provenance": "The construction has the block form\r\n\r\n[C 0 0 u 0\r\n0 C 0 u 0\r\n0 0 C u 0\r\nh1 h2 h3 0 1],\r\n\r\nwhere C is the rank-4, 7-column seed\r\n\r\nC =\r\n[1 0 0 0 1 1 1\r\n2 1 0 0 5 1 6\r\n2 0 1 0 4 3 1\r\n3 0 0 1 3 3 3],\r\n\r\nu = [1,3,6,3]^T,\r\n\r\nh1 = [2,1,6,0,1,1,4],\r\nh2 = [6,0,6,6,2,6,4],\r\nh3 = [5,2,5,2,2,0,4].\r\n\r\nThe final two columns are the distinguished pair. The matroid has rank 13 and 23 elements. Exhaustive basis enumeration and an all-pairs check give the exact global correlation constant\r\n\r\nalpha-bar = 40500/33827,\r\n\r\nattained by the final two columns.", "verifier_version": "2.0.0", "submitter": "Jeewon Kim", "created_at": "2026-08-09 14:07:17", "titles": [], "current": false }, { "id": 17, "name": "A variant of #16", "n": 23, "rank": 13, "num_bases": 439775, "alpha": { "numerator": 63, "denominator": 52, "value": 1.2115384615384615 }, "max_pairs": [ { "i": 21, "j": 22, "counts": { "containBoth": 128625, "avoidBoth": 77175, "onlyI": 191100, "onlyJ": 42875 } } ], "max_counts": { "containBoth": 128625, "avoidBoth": 77175, "onlyI": 191100, "onlyJ": 42875 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0 ], [ 0, 1, 0, 0, 4, 1, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 1, 0, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0 ], [ 0, 0, 0, 1, 4, 3, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 4, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 4, 1, 4, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 0, 0, 0, 0, 0, 0, 0, 4, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 6, 0, 0, 0, 0, 0, 0, 0, 6, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 6, 3, 1, 4, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 4, 1, 4, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 3, 4, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 3, 6, 6, 0 ], [ 0, 0, 0, 0, 1, 3, 4, 0, 0, 0, 0, 1, 3, 4, 0, 0, 0, 0, 1, 3, 4, 0, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "AI did most of the computations.", "ai_chat_url": null }, "citation": null, "provenance": "The construction has the block form\r\n\r\n[C 0 0 u 0\r\n0 C 0 u 0\r\n0 0 C u 0\r\nh h h 0 1].\r\n\r\nThe inspiration comes from #16.", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 17:25:10", "titles": [ "7" ], "current": true }, { "id": 20, "name": "Asymmetric 6-8-8 three-block affine lift over GF(4)", "n": 24, "rank": 13, "num_bases": 160704, "alpha": { "numerator": 875, "denominator": 744, "value": 1.1760752688172043 }, "max_pairs": [ { "i": 22, "j": 23, "counts": { "containBoth": 37800, "avoidBoth": 37800, "onlyI": 66960, "onlyJ": 18144 } } ], "max_counts": { "containBoth": 37800, "avoidBoth": 37800, "onlyI": 66960, "onlyJ": 18144 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [ 0, 1 ], 0 ], [ 0, 0, 0, 1, 1, [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, [ 1, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, [ 0, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, [ 0, 1 ], [ 0, 1 ], [ 0, 1 ], [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, [ 1, 1 ], [ 1, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, [ 1, 1 ], 1, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, [ 1, 1 ], 1, 0, [ 1, 1 ], [ 0, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, [ 1, 1 ], [ 1, 1 ], [ 0, 1 ], 0, 0, 0 ], [ 0, 0, [ 0, 1 ], 0, 0, [ 1, 1 ], 0, 0, 0, [ 1, 1 ], 0, [ 0, 1 ], 0, [ 1, 1 ], 0, 0, [ 0, 1 ], 0, 0, 0, 0, [ 0, 1 ], 0, 1 ] ], "field": "4", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08-09", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "The user proposed truncating one block and allowing asymmetric blocks.\r\nChatGPT derived the local block-count factorization, performed the\r\nsymmetry-reduced finite-geometry search, and exhaustively enumerated\r\nall 160,704 bases and all 276 element pairs for the final matrix.", "ai_chat_url": null }, "citation": null, "provenance": "Start from the idea of matroid #14 (three affine-lift blocks). Allow one block to have fewer columns. Search directly over the smaller block.\r\n\r\nThe matrix is of the form \r\n\r\n[ C1 0 0 u1 0 ]\r\n[ 0 C2 0 u2 0 ]\r\n[ 0 0 C3 u3 0 ]\r\n[ h1 h2 h3 0 1 ]\r\n\r\nwhere \r\nC₁ =\r\n[\r\n [1,0,0,0,0,0]\r\n [0,1,0,0,0,0]\r\n [0,0,1,0,1,1]\r\n [0,0,0,1,1,a]\r\n]\r\nC₂ =\r\n[\r\n [1,0,0,0,1,1,0,1]\r\n [0,1,0,0,0,1,1,a]\r\n [0,0,1,0,a,a,a,a]\r\n [0,0,0,1,0,1,a+1,a+1]\r\n]\r\nC₃ =\r\n[\r\n [1,0,0,0,1,0,1,0]\r\n [0,1,0,0,a+1,1,0,1]\r\n [0,0,1,0,a+1,1,0,a+1]\r\n [0,0,0,1,a+1,a+1,a,0]\r\n]\r\nu₁ =\r\n[\r\n 0\r\n 0\r\n a\r\n 1\r\n]\r\nu₂ =\r\n[\r\n a+1\r\n a\r\n 1\r\n 1\r\n]\r\nu₃ =\r\n[\r\n 0\r\n 1\r\n a\r\n 0\r\n]\r\nh₁ = [0,0,a,0,0,a+1]\r\nh₂ = [0,0,0,a+1,0,a,0,a+1]\r\nh₃ = [0,0,a,0,0,0,0,a]", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 18:56:27", "titles": [ "4" ], "current": true }, { "id": 14, "name": "Three-block sparse affine-lift GF(4) record", "n": 25, "rank": 13, "num_bases": 661932, "alpha": { "numerator": 2915, "denominator": 2484, "value": 1.173510466988728 }, "max_pairs": [ { "i": 23, "j": 24, "counts": { "containBoth": 160380, "avoidBoth": 154548, "onlyI": 268272, "onlyJ": 78732 } } ], "max_counts": { "containBoth": 160380, "avoidBoth": 154548, "onlyI": 268272, "onlyJ": 78732 }, "representation_type": "matrix", "representation": [ [ 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 1, 0, 0, [ 1, 1 ], 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 1, 0, [ 1, 1 ], 1, 0, [ 1, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [ 0, 1 ], 0 ], [ 0, 0, 1, [ 1, 1 ], [ 1, 1 ], [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, [ 1, 1 ], 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, [ 1, 1 ], 1, 0, [ 1, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, [ 0, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, [ 1, 1 ], [ 1, 1 ], [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, [ 1, 1 ], 1, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, [ 1, 1 ], 1, 0, [ 1, 1 ], [ 0, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, [ 1, 1 ], [ 1, 1 ], [ 0, 1 ], 0, 0, 0 ], [ 0, 1, [ 1, 1 ], 1, 1, 0, 0, 1, 0, 0, [ 1, 1 ], 0, [ 1, 1 ], 0, [ 0, 1 ], [ 0, 1 ], [ 0, 1 ], 1, 1, 0, 1, 0, [ 1, 1 ], 0, 1 ] ], "field": "4", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08-09", "discoverers": "Jeewon Kim", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "AI performed the finite-geometry construction search, exact finite-field computations, and exhaustive all-bases/all-pairs verification. The original search produced the 26-element 27/23 example; a one-column deletion was subsequently checked to obtain this verifier-compatible 25-element record.", "ai_chat_url": "https://chatgpt.com/share/6a78818b-8e8c-83ee-a900-83f231ef89ec" }, "citation": null, "provenance": "Let GF(4) = {0,1,a,a+1}, where a^2+a+1=0. The construction has the block form\r\n\r\n[C^- 0 0 u 0\r\n0 C 0 u 0\r\n0 0 C u 0\r\nh1 h2 h3 0 1],\r\n\r\nwhere C is the rank-4, 8-column seed\r\n\r\nC =\r\n[1 0 0 0 1 0 1 0\r\n0 1 0 0 a+1 1 0 1\r\n0 0 1 0 a+1 1 0 a+1\r\n0 0 0 1 a+1 a+1 a 0],\r\n\r\nC^- is C with its first column deleted,\r\n\r\nu = [0,1,a,0]^T,\r\n\r\nh1 = [0,1,a+1,1,1,0,0],\r\nh2 = [1,0,0,a+1,0,a+1,0,a],\r\nh3 = [a,a,1,1,0,1,0,a+1].\r\n\r\nThe final two columns are the distinguished pair. The matroid has rank 13 and 25 elements. Exhaustive basis enumeration and an all-pairs check give the exact global correlation constant\r\n\r\nalpha-bar = 2915/2484,\r\n\r\nattained uniquely by the final two columns.", "verifier_version": "2.0.0", "submitter": "Jeewon Kim", "created_at": "2026-08-09 14:01:01", "titles": [], "current": false }, { "id": 18, "name": "GF(5) three-block lift with two parallel replications", "n": 25, "rank": 13, "num_bases": 560422, "alpha": { "numerator": 1045392, "denominator": 880397, "value": 1.1874097708193008 }, "max_pairs": [ { "i": 21, "j": 22, "counts": { "containBoth": 132176, "avoidBoth": 122496, "onlyI": 251542, "onlyJ": 54208 } } ], "max_counts": { "containBoth": 132176, "avoidBoth": 122496, "onlyI": 251542, "onlyJ": 54208 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 1, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0 ], [ 0, 0, 1, 0, 3, 4, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0 ], [ 0, 0, 0, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 1 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 4, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 3, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 3, 0, 2, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 3, 4, 1, 2, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 3, 3, 3, 0, 0, 0 ], [ 0, 3, 1, 3, 1, 1, 3, 4, 3, 3, 2, 0, 3, 2, 3, 3, 1, 0, 2, 3, 3, 0, 1, 3, 3 ] ], "field": "5", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08-09", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "ChatGPT started from the existing 23-element GF(5) three-block record construction (#15) and searched for improvements obtained by adding parallel copies of columns.", "ai_chat_url": null }, "citation": null, "provenance": "This matroid is obtained from the construction in #15 by adding parallel copies of columns 2 and 9.", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 18:34:38", "titles": [ "5" ], "current": true }, { "id": 6, "name": "Double-extension sample over GF(7)", "n": 26, "rank": 9, "num_bases": 1518400, "alpha": { "numerator": 280043, "denominator": 243256, "value": 1.1512275134015193 }, "max_pairs": [ { "i": 24, "j": 25, "counts": { "containBoth": 174086, "avoidBoth": 654401, "onlyI": 486512, "onlyJ": 203401 } } ], "max_counts": { "containBoth": 174086, "avoidBoth": 654401, "onlyI": 486512, "onlyJ": 203401 }, "representation_type": "matrix", "representation": [ [ 5, 3, 2, 4, 5, 3, 2, 3, 3, 3, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 4, 1, 6, 1, 6, 6, 2, 1, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 5, 2, 2, 3, 0, 1, 3, 1, 4, 1, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 2, 4, 5, 3, 2, 3, 3, 3, 2, 4, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 6, 1, 6, 6, 2, 1, 0, 4, 4, 4, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 2, 3, 0, 1, 3, 1, 4, 1, 4, 2, 0, 0 ], [ 1, 3, 6, 0, 5, 1, 1, 0, 5, 4, 4, 4, 5, 3, 6, 3, 4, 5, 0, 6, 2, 2, 2, 1, 3, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08", "discoverers": "Chris Eur", "ai_used": true, "ai_model": "ChatGPT 5.6 Sol Ultra", "ai_role": "Found by prompting it to look at the website and beat one of the listed challenges.", "ai_chat_url": null }, "citation": null, "provenance": "Sample challenge submission by Chris Eur, August 2026: a double-(co)-extended self-direct-sum over GF(7) with alpha = 280043/243256 > 8/7, strictly beating the GF(7) field record of 8/7. Discovered by prompting ChatGPT 5.6 Sol Ultra to look at the website and beat one of the listed challenges. The maximizing pair (24, 25) is the free extension and free coextension of the construction. Submitted on Chris's behalf.", "verifier_version": "2.0.0", "submitter": "David Renshaw", "created_at": "2026-08-05 17:26:29", "titles": [], "current": false }, { "id": 10, "name": "A variant of #6", "n": 26, "rank": 9, "num_bases": 1520989, "alpha": { "numerator": 561630, "denominator": 487297, "value": 1.1525414685499809 }, "max_pairs": [ { "i": 24, "j": 25, "counts": { "containBoth": 174086, "avoidBoth": 656205, "onlyI": 487297, "onlyJ": 203401 } } ], "max_counts": { "containBoth": 174086, "avoidBoth": 656205, "onlyI": 487297, "onlyJ": 203401 }, "representation_type": "matrix", "representation": [ [ 5, 3, 2, 4, 5, 3, 2, 3, 3, 3, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 4, 1, 6, 1, 6, 6, 2, 1, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 5, 2, 2, 3, 0, 1, 3, 1, 4, 1, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 2, 4, 5, 3, 2, 3, 3, 3, 2, 4, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 6, 1, 6, 6, 2, 1, 0, 4, 4, 4, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 2, 3, 0, 1, 3, 1, 4, 1, 4, 2, 0, 0 ], [ 0, 0, 0, 0, 6, 4, 5, 2, 1, 0, 2, 5, 0, 0, 0, 0, 4, 3, 3, 0, 2, 2, 5, 1, 5, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "ChatGPT 5.6 Sol", "ai_role": "AI found it by itself by responding to my prompt: \"try to construct Matroids for GF(n) to beat the record\"", "ai_chat_url": null }, "citation": null, "provenance": "Only the last row of the matrix is different the one in #6.", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 05:45:32", "titles": [], "current": false }, { "id": 11, "name": "Optimized version of #6", "n": 26, "rank": 9, "num_bases": 1547573, "alpha": { "numerator": 98252, "denominator": 84369, "value": 1.1645509606609064 }, "max_pairs": [ { "i": 24, "j": 25, "counts": { "containBoth": 183106, "avoidBoth": 654852, "onlyI": 506214, "onlyJ": 203401 } } ], "max_counts": { "containBoth": 183106, "avoidBoth": 654852, "onlyI": 506214, "onlyJ": 203401 }, "representation_type": "matrix", "representation": [ [ 5, 3, 2, 4, 5, 3, 2, 3, 3, 3, 2, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 4, 1, 6, 1, 6, 6, 2, 1, 0, 4, 4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ], [ 5, 2, 2, 3, 0, 1, 3, 1, 4, 1, 4, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 3, 2, 4, 5, 3, 2, 3, 3, 3, 2, 4, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 1, 6, 1, 6, 6, 2, 1, 0, 4, 4, 4, 2, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 2, 3, 0, 1, 3, 1, 4, 1, 4, 2, 1, 0 ], [ 0, 0, 0, 0, 1, 2, 6, 0, 2, 3, 5, 1, 0, 0, 0, 0, 6, 5, 1, 0, 5, 4, 2, 6, 0, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "AI found the pattern described in the notes completely by itself.", "ai_chat_url": null }, "citation": null, "provenance": "Write the matrix as\r\n [C 0 u 0; 0 C u 0; x y gamma 1].\r\nHere C is a rank-4 matroid K on 12 elements, column 24 is represented by [u,u,gamma], and column 25 is represented by [0,0,1].\r\n\r\nGPT claims that for this fixed C, this construction is optimized by taking\r\nu = [1,0,2,1], x = [0,0,0,0,1,2,6,0,2,3,5,1], y = -x, and gamma = 0.", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 06:55:20", "titles": [], "current": false }, { "id": 12, "name": "A variant of #11", "n": 26, "rank": 9, "num_bases": 1478356, "alpha": { "numerator": 147888, "denominator": 125779, "value": 1.1757765604751191 }, "max_pairs": [ { "i": 24, "j": 25, "counts": { "containBoth": 178880, "avoidBoth": 611460, "onlyI": 503116, "onlyJ": 184900 } } ], "max_counts": { "containBoth": 178880, "avoidBoth": 611460, "onlyI": 503116, "onlyJ": 184900 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 5, 2, 1, 3, 5, 6, 6, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0 ], [ 0, 0, 1, 0, 5, 6, 2, 3, 4, 4, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0 ], [ 0, 0, 0, 1, 6, 3, 1, 4, 4, 4, 3, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 5, 2, 1, 3, 5, 6, 6, 5, 3, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 5, 6, 2, 3, 4, 4, 0, 1, 6, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 3, 1, 4, 4, 4, 3, 6, 2, 0 ], [ 0, 0, 0, 0, 1, 3, 5, 3, 2, 6, 0, 2, 0, 0, 0, 0, 6, 4, 2, 4, 5, 1, 0, 5, 0, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "AI did the intensive computations.", "ai_chat_url": null }, "citation": null, "provenance": "Write the matrix as\r\n [C 0 u 0; 0 C u 0; h -h 0 1], which also appears in #11.\r\n\r\nGPT claims for this new C, this construction is optimized by taking u=[1,3,6,2], h=[0,0,0,0,1,3,5,3,2,6,0,2].", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 08:00:25", "titles": [], "current": false }, { "id": 13, "name": "[C 0 u 0; 0 C u 0; h h 0 1]", "n": 26, "rank": 9, "num_bases": 1131928, "alpha": { "numerator": 55752, "denominator": 48391, "value": 1.1521150627182741 }, "max_pairs": [ { "i": 24, "j": 25, "counts": { "containBoth": 139840, "avoidBoth": 460560, "onlyI": 387128, "onlyJ": 144400 } } ], "max_counts": { "containBoth": 139840, "avoidBoth": 460560, "onlyI": 387128, "onlyJ": 144400 }, "representation_type": "matrix", "representation": [ [ 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 1, 0, [ 1, 1 ], 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [ 1, 1 ], 0 ], [ 0, 0, 1, 0, 0, 1, [ 1, 1 ], [ 1, 1 ], 1, [ 0, 1 ], 1, [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 1, 1, 0, 1, 1, 1, [ 0, 1 ], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, [ 1, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 1, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, [ 1, 1 ], 0, 1, 0, 1, 1, [ 1, 1 ], 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, [ 1, 1 ], [ 1, 1 ], 1, [ 0, 1 ], 1, [ 0, 1 ], 1, 0 ], [ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, [ 0, 1 ], 0, 0, [ 1, 1 ], 0 ], [ 0, 0, 0, 0, 1, [ 0, 1 ], [ 0, 1 ], [ 1, 1 ], [ 0, 1 ], 0, 0, [ 1, 1 ], 0, 0, 0, 0, 1, [ 0, 1 ], [ 0, 1 ], [ 1, 1 ], [ 0, 1 ], 0, 0, [ 1, 1 ], 0, 1 ] ], "field": "4", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2026-08", "discoverers": "Yongxi (Aaron) Lin", "ai_used": true, "ai_model": "GPT-5.6 Sol", "ai_role": "AI uses the matrix that appears in #12.", "ai_chat_url": null }, "citation": null, "provenance": "The construction has the form\r\n [C 0 u 0; 0 C u 0; h h 0 1]\r\nwith a rank-4, 12-column seed C.\r\nu = [1,3,1,3]\r\nh = [1,2,2,3,2,0,0,3]", "verifier_version": "2.0.0", "submitter": "Yongxi (Aaron) Lin", "created_at": "2026-08-09 08:11:58", "titles": [], "current": false }, { "id": 7, "name": "HSW finite-field spike M_7^5", "n": 30, "rank": 5, "num_bases": 46991, "alpha": { "numerator": 8, "denominator": 7, "value": 1.1428571428571428 }, "max_pairs": [ { "i": 0, "j": 1, "counts": { "containBoth": 1372, "avoidBoth": 28812, "onlyI": 2401, "onlyJ": 14406 } } ], "max_counts": { "containBoth": 1372, "avoidBoth": 28812, "onlyI": 2401, "onlyJ": 14406 }, "representation_type": "matrix", "representation": [ [ 1, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0, 1, 2, 3, 4, 5, 6, 0 ], [ 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0 ], [ 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1 ] ], "field": "7", "cert_field": null, "cert_matrix": null, "bases": null, "discovery": { "date": "2018-06", "discoverers": "June Huh, Benjamin Schröter, Botong Wang", "ai_used": false, "ai_model": null, "ai_role": null, "ai_chat_url": null }, "citation": "arXiv:1806.02675, Definition 19 (JEMS 24 (2022), 1335–1351)", "provenance": "The finite-field spike M_7^5 of Huh, Schroter, and Wang (Correlation bounds for fields and matroids, arXiv:1806.02675, Definition 19; discovered June 2018, no AI use reported in that record). Columns over GF(7): the tip e1, the vector e2+e3+e4+e5, and k*e1+e_m for m=2..5, k=1..p. Submitted from the literature; completes the four spikes M_p^5 for p in {2,3,5,7}.", "verifier_version": "2.0.0", "submitter": "David Renshaw", "created_at": "2026-08-05 17:26:47", "titles": [], "current": false } ] }