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Also, for a subset $T$ of $\\mathbb{N} = \\{1,2,\\cdots \\}$, a finite subset $Y\\subset S^{n-1}$ is called a spherical design of harmonic index $T,$ if $\\sum_{\\mathbf{x}\\in Y}f(\\mathbf{x})=0$ is satisfied for all real homogeneous harmonic polynomials $f(x_1,\\ldots,x_n)$ of degree $k$ with $k\\in T$.\n  In the present p"},"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":"1507.05373","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-20T02:53:54Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"fa69ab05b606e7d2a59b0d363322103ed203065575177128f55f54ff7d519acb","abstract_canon_sha256":"93b1b3d2720dc61c25fd6c79eefd1cb681f5dc0c915519341a94d01097ae97bd"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:32.919857Z","signature_b64":"i8trXyxCU0mqmT5D+ftyunAgRQAGZPYokvHaSX0rV0w1DxFTpoKqAt0gpLqc+W5woRbcXXRuffo4nGapEFHSAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30d8813f9213e148aa08adda00c2faa3b98187b595229cd7f6ff178741242a0c","last_reissued_at":"2026-05-18T01:36:32.919222Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:32.919222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"More on spherical designs of harmonic index $t$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Eiichi Bannai, Etsuko Bannai, Kyoung-Tark Kim, Wei-Hsuan Yu, Yan Zhu","submitted_at":"2015-07-20T02:53:54Z","abstract_excerpt":"A finite subset $Y$ on the unit sphere $S^{n-1} \\subseteq \\mathbb{R}^n$ is called a spherical design of harmonic index $t$, if the following condition is satisfied: $\\sum_{\\mathbf{x}\\in Y}f(\\mathbf{x})=0$ for all real homogeneous harmonic polynomials $f(x_1,\\ldots,x_n)$ of degree $t$. 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