{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,27]],"date-time":"2025-10-27T08:34:50Z","timestamp":1761554090992,"version":"build-2065373602"},"reference-count":22,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2024,6,7]],"date-time":"2024-06-07T00:00:00Z","timestamp":1717718400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,6,7]],"date-time":"2024-06-07T00:00:00Z","timestamp":1717718400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2025,11]]},"abstract":"Abstract<\/jats:title>\n \n Let\n \n \n $${{\\mathbb {K}}}$$<\/jats:tex-math>\n \n K<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be any field, let\n \n \n $$X\\subset {\\mathbb P}^{k-1}$$<\/jats:tex-math>\n \n \n X<\/mml:mi>\n \u2282<\/mml:mo>\n \n \n P<\/mml:mi>\n <\/mml:mrow>\n \n k<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be a set of\n \n \n $$n$$<\/jats:tex-math>\n \n n<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n distinct\n \n \n $${{\\mathbb {K}}}$$<\/jats:tex-math>\n \n K<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -rational points, and let\n \n \n $$a\\ge 1$$<\/jats:tex-math>\n \n \n a<\/mml:mi>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n be an integer. In this paper we find lower bounds for the minimum distance\n \n \n $$d(X)_a$$<\/jats:tex-math>\n \n \n d<\/mml:mi>\n \n \n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n a<\/mml:mi>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n of the evaluation code of order\n \n \n $$a$$<\/jats:tex-math>\n \n a<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n associated to\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . The first results use\n \n \n $$\\alpha (X)$$<\/jats:tex-math>\n \n \n \u03b1<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the initial degree of the defining ideal of\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , and the bounds are true for any set\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . In another result we use\n \n \n $$s(X)$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n (<\/mml:mo>\n X<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , the minimum socle degree, to find a lower bound for the case when\n \n \n $$X$$<\/jats:tex-math>\n \n X<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is in general linear position. In both situations we improve and generalize known results.\n <\/jats:p>","DOI":"10.1007\/s00200-024-00659-1","type":"journal-article","created":{"date-parts":[[2024,6,7]],"date-time":"2024-06-07T03:02:06Z","timestamp":1717729326000},"page":"1213-1235","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Geometry of the minimum distance"],"prefix":"10.1007","volume":"36","author":[{"given":"John","family":"Pawlina","sequence":"first","affiliation":[]},{"given":"\u015etefan O.","family":"Toh\u01ceneanu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,6,7]]},"reference":[{"key":"659_CR1","doi-asserted-by":"publisher","first-page":"135","DOI":"10.1007\/s00200-006-0012-y","volume":"17","author":"E Ballico","year":"2006","unstructured":"Ballico, E., Fontanari, C.: The Horace method for error-correcting codes. Appl. Algebra Eng. Commun. 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