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Commun."],"published-print":{"date-parts":[[2026,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We prove that for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n&gt;1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    the map\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\chi :\\mathbb {F}_q^n \\rightarrow \\mathbb {F}_q^n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>\u03c7<\/mml:mi>\n                            <mml:mo>:<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msubsup>\n                            <mml:mo>\u2192<\/mml:mo>\n                            <mml:msubsup>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mi>q<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msubsup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , defined by\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$y=\\chi (x)$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mi>\u03c7<\/mml:mi>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mi>x<\/mml:mi>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    with\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$y_i = x_i + x_{i+2}\\cdot (1+x_{i+1})$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>x<\/mml:mi>\n                              <mml:mrow>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>\u00b7<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>x<\/mml:mi>\n                                <mml:mrow>\n                                  <mml:mi>i<\/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:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$1\\le i \\le n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , is bijective if and only if\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$q=2$$<\/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>2<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:italic>n<\/jats:italic>\n                    is odd, as it was conjectured by Schoone and Daemen in 2024.\n                  <\/jats:p>","DOI":"10.1007\/s12095-025-00786-7","type":"journal-article","created":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T00:05:51Z","timestamp":1740096351000},"page":"97-105","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["On the bijectivity of the map $$\\chi 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