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Codes Cryptogr."],"published-print":{"date-parts":[[2023,7]]},"abstract":"Abstract<\/jats:title>In this paper we settle the question of whether a finite-dimensional vector space $${{\\mathcal {V}}}$$<\/jats:tex-math>\n V<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> over $${\\mathbb {F}}_p,$$<\/jats:tex-math>\n \n \n F<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> with p<\/jats:italic> an odd prime, and the family of all the k<\/jats:italic>-sets of elements of $${\\mathcal {V}}$$<\/jats:tex-math>\n V<\/mml:mi>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> summing up to a given element x<\/jats:italic>,\u00a0 form a 1-$$(v,k,\\lambda _1)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n v<\/mml:mi>\n ,<\/mml:mo>\n k<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> or a 2-$$(v,k,\\lambda _2)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n v<\/mml:mi>\n ,<\/mml:mo>\n k<\/mml:mi>\n ,<\/mml:mo>\n \n \u03bb<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> block design, and, in either case, we find a closed form for $$\\lambda _i$$<\/jats:tex-math>\n \n \u03bb<\/mml:mi>\n i<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and characterize the automorphism group. The question is discussed also in the case where the elements of the k<\/jats:italic>-sets are required to be all nonzero, as the two cases happen to be intrinsically inseparable. The \u201ctwin case\u201d $$p=2,$$<\/jats:tex-math>\n \n p<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which has strict connections with coding theory, was completely discussed in a recent paper by G. Falcone and the present author.<\/jats:p>","DOI":"10.1007\/s10623-023-01213-9","type":"journal-article","created":{"date-parts":[[2023,4,19]],"date-time":"2023-04-19T14:03:27Z","timestamp":1681913007000},"page":"2585-2603","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Subset sums and block designs in a finite vector space"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8674-5841","authenticated-orcid":false,"given":"Marco","family":"Pavone","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2023,4,19]]},"reference":[{"key":"1213_CR1","unstructured":"Bassalygo L.A., Zinov\u2019ev V.A.: A note on uniformly-packed codes (in Russian). Probl. 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