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Codes Cryptogr."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Datta and Johnsen (Des Codes Cryptogr 91:747\u2013761, 2023) introduced a new family of evaluation codes in an affine space of dimension <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\ge 2$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> over a finite field <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\mathbb {F}}_q$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:mi>q<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> where linear combinations of elementary symmetric polynomials are evaluated on the set of all points with pairwise distinct coordinates. In this paper, we propose a generalization by taking low dimensional linear systems of symmetric polynomials. Computation for small values of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$q=7,9$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>7<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>9<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> shows that carefully chosen generalized Datta\u2013Johnsen codes <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\left[ \\frac{1}{2}q(q-1),3,d\\right] $$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mfenced>\n                    <mml:mfrac>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mfrac>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>q<\/mml:mi>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                  <\/mml:mfenced>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> have minimum distance <jats:italic>d<\/jats:italic> equal to the optimal value minus 1.<\/jats:p>","DOI":"10.1007\/s10623-025-01637-5","type":"journal-article","created":{"date-parts":[[2025,5,12]],"date-time":"2025-05-12T10:58:00Z","timestamp":1747047480000},"page":"3361-3373","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Evaluation codes arising from symmetric polynomials"],"prefix":"10.1007","volume":"93","author":[{"given":"Barbara","family":"Gatti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G\u00e1bor","family":"Korchm\u00e1ros","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G\u00e1bor P.","family":"Nagy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Vincenzo","family":"Pallozzi Lavorante","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gioia","family":"Schulte","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,5,12]]},"reference":[{"key":"1637_CR1","doi-asserted-by":"publisher","first-page":"603","DOI":"10.1515\/advgeom.2011.022","volume":"11","author":"V Abatangelo","year":"2011","unstructured":"Abatangelo V., Fisher C.J., Korchm\u00e1ros G., Larato B.: On the mutual position of two irreducible conics in $$PG(2, q)$$, $$q$$ odd. 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