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In this note, by analyzing the weight of $\\sigma_{n, 2^t}$ and $\\sigma_{n, d}$, we prove that ${\\rm wt}(\\sigma_{n, d})<2^{n-1}$ holds in most cases, and so does the conjecture. According to the remainder of modulo 4, we also consider the weight of $\\sigma_{n, d}$ from "},"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":"1203.1418","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2012-03-07T09:47:08Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"5f8e6c7015fcd003594f5f492498884d7b010427e3e7ea773a91cbb6d8fc8208","abstract_canon_sha256":"c5ddeaf7e5bb754ed046b823e85aadab7e0212c7892924ad3197eeffeeabe6db"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:21:17.229335Z","signature_b64":"QB7sx3/zAS4x9Tt7ugpHgntiEJZRyzkL6NDpX9fGpGMB3LFUC67RuRqbLtFc9vFD2QF4hvHtADwzWrSzZd6vAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d916e77438407e3c4f40e0fb0ebcc7dceebb715e1176f1d321645fa0af9520bb","last_reissued_at":"2026-05-18T02:21:17.228718Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:21:17.228718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Note on a Conjecture for Balanced Elementary Symmetric Boolean Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Alexander Pott, Wei Su, Xiaohu Tang","submitted_at":"2012-03-07T09:47:08Z","abstract_excerpt":"In 2008, Cusick {\\it et al.} conjectured that certain elementary symmetric Boolean functions of the form $\\sigma_{2^{t+1}l-1, 2^t}$ are the only nonlinear balanced ones, where $t$, $l$ are any positive integers, and $\\sigma_{n,d}=\\bigoplus_{1\\le i_1<...<i_d\\le n}x_{i_1}x_{i_2}...x_{i_d}$ for positive integers $n$, $1\\le d\\le n$. In this note, by analyzing the weight of $\\sigma_{n, 2^t}$ and $\\sigma_{n, d}$, we prove that ${\\rm wt}(\\sigma_{n, d})<2^{n-1}$ holds in most cases, and so does the conjecture. 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