{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T00:05:23Z","timestamp":1706659523085},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T00:00:00Z","timestamp":1674950400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T00:00:00Z","timestamp":1674950400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Universit\u00e0 degli Studi di Roma La Sapienza"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2023,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb K}$$<\/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 the Galois field <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb F}_{q^t}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>F<\/mml:mi>\n                    <mml:msup>\n                      <mml:mi>q<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> of order <jats:inline-formula><jats:alternatives><jats:tex-math>$$q^t, q=p^e, p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mi>q<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>q<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mi>e<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>p<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> a prime, <jats:inline-formula><jats:alternatives><jats:tex-math>$$A={{\\,\\mathrm{{Aut}}\\,}}({\\mathbb K})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>A<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mspace \/>\n                      <mml:mi>Aut<\/mml:mi>\n                      <mml:mspace \/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>K<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> be the automorphism group of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb K}$$<\/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> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\sigma }=(\\sigma _0,\\ldots , \\sigma _{d-1}) \\in A^d$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>\u03c3<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>\u03c3<\/mml:mi>\n                        <mml:mn>0<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mo>\u2026<\/mml:mo>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>\u03c3<\/mml:mi>\n                        <mml:mrow>\n                          <mml:mi>d<\/mml:mi>\n                          <mml:mo>-<\/mml:mo>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:mrow>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>A<\/mml:mi>\n                      <mml:mi>d<\/mml:mi>\n                    <\/mml:msup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$d \\ge 1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>\u2265<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. In this paper the following generalization of the Veronese map is studied: <jats:disp-formula><jats:alternatives><jats:tex-math>$$\\begin{aligned} \\nu _{d,\\varvec{\\sigma }} : \\langle v \\rangle \\in {{\\,\\mathrm{{PG}}\\,}}(n-1,{\\mathbb K}) \\longrightarrow \\langle v^{\\sigma _0} \\otimes v^{\\sigma _1} \\otimes \\cdots \\otimes v^{\\sigma _{d-1}} \\rangle \\in {{\\,\\mathrm{{PG}}\\,}}(n^d-1,{\\mathbb K}). \\end{aligned}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtable>\n                      <mml:mtr>\n                        <mml:mtd>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\u03bd<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>d<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mrow>\n                                  <mml:mi>\u03c3<\/mml:mi>\n                                <\/mml:mrow>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>\u27e8<\/mml:mo>\n                              <mml:mi>v<\/mml:mi>\n                              <mml:mo>\u27e9<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mspace \/>\n                              <mml:mi>PG<\/mml:mi>\n                              <mml:mspace \/>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mo>-<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>K<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u27f6<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>\u27e8<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>v<\/mml:mi>\n                                <mml:msub>\n                                  <mml:mi>\u03c3<\/mml:mi>\n                                  <mml:mn>0<\/mml:mn>\n                                <\/mml:msub>\n                              <\/mml:msup>\n                              <mml:mo>\u2297<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>v<\/mml:mi>\n                                <mml:msub>\n                                  <mml:mi>\u03c3<\/mml:mi>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:msub>\n                              <\/mml:msup>\n                              <mml:mo>\u2297<\/mml:mo>\n                              <mml:mo>\u22ef<\/mml:mo>\n                              <mml:mo>\u2297<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>v<\/mml:mi>\n                                <mml:msub>\n                                  <mml:mi>\u03c3<\/mml:mi>\n                                  <mml:mrow>\n                                    <mml:mi>d<\/mml:mi>\n                                    <mml:mo>-<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                              <\/mml:msup>\n                              <mml:mo>\u27e9<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mspace \/>\n                              <mml:mi>PG<\/mml:mi>\n                              <mml:mspace \/>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msup>\n                                <mml:mi>n<\/mml:mi>\n                                <mml:mi>d<\/mml:mi>\n                              <\/mml:msup>\n                              <mml:mo>-<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>K<\/mml:mi>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mtd>\n                      <\/mml:mtr>\n                    <\/mml:mtable>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:disp-formula>Its image will be called the <jats:inline-formula><jats:alternatives><jats:tex-math>$$(d,\\varvec{\\sigma })$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mi>\u03c3<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-<jats:italic>Veronese variety<\/jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal V_{d,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For <jats:inline-formula><jats:alternatives><jats:tex-math>$$d=t$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\sigma $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c3<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> a generator of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textrm{Gal}({\\mathbb F}_{q^t}|{\\mathbb F}_{q})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>Gal<\/mml:mtext>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:msup>\n                        <mml:mi>q<\/mml:mi>\n                        <mml:mi>t<\/mml:mi>\n                      <\/mml:msup>\n                    <\/mml:msub>\n                    <mml:mo>|<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mi>q<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\varvec{\\sigma }=(1,\\sigma ,\\sigma ^2,\\ldots ,\\sigma ^{t-1})$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>\u03c3<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>\u03c3<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u03c3<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mo>\u2026<\/mml:mo>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>\u03c3<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>t<\/mml:mi>\n                        <mml:mo>-<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the <jats:inline-formula><jats:alternatives><jats:tex-math>$$(t,\\varvec{\\sigma })$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mrow>\n                      <mml:mi>\u03c3<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-Veronese variety <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal V_{t,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>t<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the variety studied in [9, 11, 13]. Such a variety is the Grassmann embedding of the Desarguesian spread of <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{{PG}}\\,}}(nt-1,{\\mathbb F}_q)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace \/>\n                      <mml:mi>PG<\/mml:mi>\n                      <mml:mspace \/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                    <mml:mi>t<\/mml:mi>\n                    <mml:mo>-<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mi>q<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and it has been used to construct codes [3] and (partial) ovoids of quadrics, see [9, 12]. Here, we will show that <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal V_{d,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the Grassmann embedding of a normal rational scroll and any <jats:inline-formula><jats:alternatives><jats:tex-math>$$d+1$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> points of it are linearly independent. We give a characterization of <jats:inline-formula><jats:alternatives><jats:tex-math>$$d+2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>d<\/mml:mi>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> linearly dependent points of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal V_{d,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and for some choices of parameters, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal V_{p,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the normal rational curve; for <jats:inline-formula><jats:alternatives><jats:tex-math>$$p=2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, it can be the Segre\u2019s arc of <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{{PG}}\\,}}(3,q^t)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace \/>\n                      <mml:mi>PG<\/mml:mi>\n                      <mml:mspace \/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>q<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>; for <jats:inline-formula><jats:alternatives><jats:tex-math>$$p=3$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mn>3<\/mml:mn>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula><jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal V_{p,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>V<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> can be also a <jats:inline-formula><jats:alternatives><jats:tex-math>$$|\\mathcal V_{p,\\varvec{\\sigma }}|$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msub>\n                      <mml:mi>V<\/mml:mi>\n                      <mml:mrow>\n                        <mml:mi>p<\/mml:mi>\n                        <mml:mo>,<\/mml:mo>\n                        <mml:mrow>\n                          <mml:mi>\u03c3<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:mrow>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>|<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-track of <jats:inline-formula><jats:alternatives><jats:tex-math>$${{\\,\\mathrm{{PG}}\\,}}(5,q^t)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mrow>\n                      <mml:mspace \/>\n                      <mml:mi>PG<\/mml:mi>\n                      <mml:mspace \/>\n                    <\/mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mn>5<\/mml:mn>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>q<\/mml:mi>\n                      <mml:mi>t<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Finally, investigate the link between such points sets and a linear code <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal C}_{d,\\varvec{\\sigma }}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>C<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>\u03c3<\/mml:mi>\n                      <\/mml:mrow>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that can be associated to the variety, obtaining examples of MDS and almost MDS codes.<\/jats:p>","DOI":"10.1007\/s10623-023-01186-9","type":"journal-article","created":{"date-parts":[[2023,1,29]],"date-time":"2023-01-29T11:03:00Z","timestamp":1674990180000},"page":"1911-1921","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["$$(d,\\varvec{\\sigma })$$-Veronese variety and some applications"],"prefix":"10.1007","volume":"91","author":[{"given":"N.","family":"Durante","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"G.","family":"Longobardi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"V.","family":"Pepe","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,1,29]]},"reference":[{"key":"1186_CR1","volume-title":"Algebraic Coding Theory","author":"ER Berlekamp","year":"1968","unstructured":"Berlekamp E.R.: Algebraic Coding Theory. 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