{"paper":{"title":"Optimal $2$-D $(n\\times m,3,2,1)$-optical orthogonal codes and related equi-difference conflict avoiding codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Lidong Wang, Tao Feng, Xiaomiao Wang","submitted_at":"2018-04-12T12:31:47Z","abstract_excerpt":"This paper focuses on constructions for optimal $2$-D $(n\\times m,3,2,1)$-optical orthogonal codes with $m\\equiv 0\\ ({\\rm mod}\\ 4)$. An upper bound on the size of such codes is established. It relies heavily on the size of optimal equi-difference $1$-D $(m,3,2,1)$-optical orthogonal codes, which is closely related to optimal equi-difference conflict avoiding codes with weight $3$. The exact number of codewords of an optimal $2$-D $(n\\times m,3,2,1)$-optical orthogonal code is determined for $n=1,2$, $m\\equiv 0 \\pmod{4}$, and $n\\equiv 0 \\pmod{3}$, $m\\equiv 8 \\pmod{16}$ or $m\\equiv 32 \\pmod{64}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.04467","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}