{"paper":{"title":"Nonexistence of certain classes of generalized bent functions: Revisiting the element partition method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The element partition method establishes nonexistence of generalized bent functions of type [n, 2 p1^e1 p2^e2] and type [1, 2·3^a·7^b].","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Shi Ying, Yingpu Deng","submitted_at":"2026-05-13T14:00:15Z","abstract_excerpt":"We obtain new nonexistence results of two classes of generalized bent functions from $\\mathbb{Z}_{q}^{n}$ to $\\mathbb{Z}_{q}$ (called type $[n,q]$). The first class of results is based on applying the element partition method to the results of Feng and Feng and Liu, where $q=2 p_1^{e_1} p_{2}^{e_2}$, $p_1$ and $p_2$ are two primes. For the second class, we extend the idea of the element partition method and prove the nonexistence of generalized bent functions of type $[1,2 \\cdot 3^{a} \\cdot 7^{b}]$, where $a,b \\in \\mathbb{Z}_{>0}$."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We obtain new nonexistence results of two classes of generalized bent functions from Z_q^n to Z_q (called type [n,q]). The first class ... where q=2 p1^e1 p2^e2 ... For the second class, we extend the idea of the element partition method and prove the nonexistence of generalized bent functions of type [1,2·3^a·7^b].","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the element partition method, when applied to the cited results of Feng et al., produces a contradiction for the stated parameter ranges without hidden assumptions on the character sums or on the distribution of elements in the ring Z_q.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Nonexistence is shown for generalized bent functions from Z_q^n to Z_q when q = 2 p1^e1 p2^e2 and when n=1 and q=2·3^a·7^b.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The element partition method establishes nonexistence of generalized bent functions of type [n, 2 p1^e1 p2^e2] and type [1, 2·3^a·7^b].","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"c3941d766106b55d258cbb5eecebcb96ba5e302bf47fd8bd4201772d8ce66445"},"source":{"id":"2605.13558","kind":"arxiv","version":1},"verdict":{"id":"cd68116b-a7de-45e6-b0e3-650963ba173e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T17:55:55.607952Z","strongest_claim":"We obtain new nonexistence results of two classes of generalized bent functions from Z_q^n to Z_q (called type [n,q]). The first class ... where q=2 p1^e1 p2^e2 ... For the second class, we extend the idea of the element partition method and prove the nonexistence of generalized bent functions of type [1,2·3^a·7^b].","one_line_summary":"Nonexistence is shown for generalized bent functions from Z_q^n to Z_q when q = 2 p1^e1 p2^e2 and when n=1 and q=2·3^a·7^b.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the element partition method, when applied to the cited results of Feng et al., produces a contradiction for the stated parameter ranges without hidden assumptions on the character sums or on the distribution of elements in the ring Z_q.","pith_extraction_headline":"The element partition method establishes nonexistence of generalized bent functions of type [n, 2 p1^e1 p2^e2] and type [1, 2·3^a·7^b]."},"references":{"count":30,"sample":[{"doi":"","year":1997,"title":"Fermat quotients for composite moduli.J","work_id":"6ddada85-0053-4ebd-9a48-0d53c4d5b85b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1998,"title":"Bruce C. Berndt, Ronald J. Evans, and Kenneth S. Williams.Gauss and Jacobi sums. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wi- ley & Sons, Inc., New York, 1998. A Wil","work_id":"65d91714-155b-4edd-bf12-6f1dea84a6fe","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2016,"title":"Four decades of research on bent functions.Des","work_id":"2185a8ed-3d95-4dca-a76b-c21a34cd1f91","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2011,"title":"Dorais and Dominic Klyve","work_id":"de0056da-3a23-4bea-a5b5-41ba69ab260a","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"Non-existence of some generalized bent functions","work_id":"172b42e2-c341-4b1e-8223-87e0fe01737f","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"345c1c7cdf512103469ac216e644826ba74e0377a6c694507936d311b9717162","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}