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We investigate the dimensions of the binary codes derived from $U_\\theta$.\n  By using Kloosterman sums, we obtain a new lower bound on the aforementioned dimensions which improves the result obtained by Leung and Xiang in 2009. In particular, for $q=3^m$, this new lower bound equals $\\frac{2}{3}(q^3+q^2-2q)-1$ for even $m$ and $\\frac{2}{3}(q^3+q^2+q)-"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1510.03340","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-12T15:37:48Z","cross_cats_sorted":[],"title_canon_sha256":"8a0d36f75190f45151b3d702ad99f996fde014eaaf49d3cc5622d0158e236768","abstract_canon_sha256":"bf2f908529fd92449aa7a405772543aebd3f2e60c3bf4049d9db45517327edd6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:32.155878Z","signature_b64":"Zby3vv8aO8qtcz+w/F6jod6FWD8PI0bce2qphY8LoL7cUQMxBZcsB5Pa10QvlUAdrR23EWhsQRZvSlY9oUcUAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c0f4d1e7da65d172b5989d18f3631eb722c5fec4c1bcda52e997df009b90fec","last_reissued_at":"2026-05-18T01:30:32.155164Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:32.155164Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the 2-ranks of a class of unitals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Rocco Trombetti, Yue Zhou","submitted_at":"2015-10-12T15:37:48Z","abstract_excerpt":"Let $U_\\theta$ be a unital defined in a shift plane of odd order $q^2$, which are constructed recently by the authors. 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