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Codes Cryptogr."],"published-print":{"date-parts":[[2024,12]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {K}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>K<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be a discrete valued field with finite residue field. In analogy with orthogonality in the Euclidean space <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {R}}^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>R<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, there is a well-studied notion of \u201cultrametric orthogonality\u201d in <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {K}}^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>K<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper, motivated by a question of Erd\u0151s in the real case, given integers <jats:inline-formula><jats:alternatives><jats:tex-math>$$k \\ge \\ell \\ge 2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>k<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mi>\u2113<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we investigate the maximum size of a subset <jats:inline-formula><jats:alternatives><jats:tex-math>$$S \\subseteq {\\mathcal {K}}^n {\\setminus }\\{\\textbf{0}\\}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>S<\/mml:mi>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:msup>\n                      <mml:mrow>\n                        <mml:mi>K<\/mml:mi>\n                      <\/mml:mrow>\n                      <mml:mi>n<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>\\<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>{<\/mml:mo>\n                      <mml:mn>0<\/mml:mn>\n                      <mml:mo>}<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> satisfying the following property: for any <jats:inline-formula><jats:alternatives><jats:tex-math>$$E \\subseteq S$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>E<\/mml:mi>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mi>S<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of size <jats:italic>k<\/jats:italic>, there exists <jats:inline-formula><jats:alternatives><jats:tex-math>$$F \\subseteq E$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mo>\u2286<\/mml:mo>\n                    <mml:mi>E<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of size <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u2113<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> such that any two distinct vectors in <jats:italic>F<\/jats:italic> are orthogonal. Other variants of this property are also studied.<\/jats:p>","DOI":"10.1007\/s10623-024-01480-0","type":"journal-article","created":{"date-parts":[[2024,8,16]],"date-time":"2024-08-16T11:03:03Z","timestamp":1723806183000},"page":"4035-4055","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On the maximum size of ultrametric orthogonal sets over discrete valued fields"],"prefix":"10.1007","volume":"92","author":[{"given":"Noy Soffer","family":"Aranov","sequence":"first","affiliation":[]},{"given":"Angelot","family":"Behajaina","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2024,8,16]]},"reference":[{"key":"1480_CR1","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1016\/j.ffa.2015.09.009","volume":"37","author":"O Ahmadi","year":"2016","unstructured":"Ahmadi O., Mohammadian A.: Sets with many pairs of orthogonal vectors over finite fields. 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