{"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. According to the remainder of modulo 4, we also consider the weight of $\\sigma_{n, d}$ from "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1203.1418","kind":"arxiv","version":2},"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"}