{"paper":{"title":"Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Univariate bicycle codes reduce quantum LDPC design to a single polynomial while providing an explicit basis for logical operators and upper bounds on distance.","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Hessam Mahdavifar, Sheida Rabeti","submitted_at":"2026-05-13T22:52:01Z","abstract_excerpt":"We introduce univariate bicycle (UB) codes, a structured subclass of generalized bicycle (GB) quantum low-density parity-check (LDPC) codes obtained via a Frobenius relation. This construction reduces the code design space from a two-polynomial search in GB codes to a single-polynomial search, while preserving sparsity. We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The Frobenius relation applied to the defining polynomials preserves the sparsity, the quantum CSS structure, and the low-density parity-check property of the original generalized bicycle codes.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Univariate bicycle codes give an explicit basis for logical operators and distance upper bounds in a restricted class of quantum LDPC codes while matching the performance of less constrained generalized and bivariate bicycle codes in simulations.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Univariate bicycle codes reduce quantum LDPC design to a single polynomial while providing an explicit basis for logical operators and upper bounds on distance.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"81f6e24c47d83f2d532146eb686afc542b800bcdfc3fcce75319efe9574bb9fd"},"source":{"id":"2605.14173","kind":"arxiv","version":1},"verdict":{"id":"ba6d3c44-009a-4cf7-af4c-d6a336d0b903","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T01:38:44.733724Z","strongest_claim":"We provide an explicit algebraic characterization of the logical coset spaces by constructing a basis for the logical quotient space, yielding a complete parametrization of logical operators. Leveraging this structure, we derive upper bounds on the minimum distance by relating structured logical representatives to cycle-density properties of associated circulant matrices.","one_line_summary":"Univariate bicycle codes give an explicit basis for logical operators and distance upper bounds in a restricted class of quantum LDPC codes while matching the performance of less constrained generalized and bivariate bicycle codes in simulations.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The Frobenius relation applied to the defining polynomials preserves the sparsity, the quantum CSS structure, and the low-density parity-check property of the original generalized bicycle codes.","pith_extraction_headline":"Univariate bicycle codes reduce quantum LDPC design to a single polynomial while providing an explicit basis for logical operators and upper bounds on distance."},"references":{"count":23,"sample":[{"doi":"","year":1962,"title":"R. Gallager, “Low-density parity-check codes,”IRE Transactions on information theory, vol. 8, no. 1, pp. 21–28, 1962","work_id":"e4fcd8e9-c455-415e-b63f-fd5aa6f9b259","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"Fifteen years of quantum LDPC coding and improved decoding strategies,","work_id":"6d1032ff-1fa0-4f30-a96d-e26cb7c65ad3","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2021,"title":"Quantum low-density parity- check codes,","work_id":"fcccd828-c25e-4fb1-8187-d0d9eab61fdd","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Quantum low-density parity-check codes","work_id":"309b64a3-ed84-4a6b-ab00-a1b9d6ac53d0","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"Quantum kronecker sum-product low- density parity-check codes with finite rate,","work_id":"68a35ce6-e9e0-4c98-9de2-94e71277ffbe","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":23,"snapshot_sha256":"d8be7a1a2792c08f6d0e30f054f60247afc68d47bc404f2f0f2ce67577156d77","internal_anchors":2},"formal_canon":{"evidence_count":2,"snapshot_sha256":"b19d5ceb99460290159104bddd8349c0cad2d7c53de31c142a8fa2de01d00b3d"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}