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By Gray mapping, we obtain two infinite families of linear $p$-ary codes of respective lengths $(p^m-1)^2$ and $2(p^m-1)^2.$ When $m$ is singly-even, the first family gives five-weight codes. When $m$ is odd, and $p\\equiv 3 \\pmod{4},$ the first family yields $p$-ary two-weight codes, which are shown to be optimal by application of the Gries"},"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":"1612.00967","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-12-03T13:27:27Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"61345bff3b3c3da6c3df7130d4dd460c50f76d3c60299ca287dc0e6b36082747","abstract_canon_sha256":"3b3ffee99065650fd0cfa27197fb8e130dc762803153db89bcd2cb2b30c8bd48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:35:09.363378Z","signature_b64":"dhPh5q0zZzQ8FbIV3T0Z2TlTBs6Trmx3M+gs9fvD5M4MvSo2pAfWkPjOIcnVnHQgCBrkFR33pfhqhc0e/H2hDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"509cc6417a1572096b015c2829ccc3b34ffe02ec8373491ee74f25412957b481","last_reissued_at":"2026-05-18T00:35:09.362833Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:35:09.362833Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Two new families of two-weight codes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Minjia Shi, Patrick Sole, Yue Guan","submitted_at":"2016-12-03T13:27:27Z","abstract_excerpt":"We construct two new infinite families of trace codes of dimension $2m$, over the ring $\\mathbb{F}_p+u\\mathbb{F}_p,$ when $p$ is an odd prime. 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