{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:42:08Z","timestamp":1740148928727,"version":"3.37.3"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T00:00:00Z","timestamp":1620000000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T00:00:00Z","timestamp":1620000000000},"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":["Cryptogr. Commun."],"published-print":{"date-parts":[[2021,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this work, we study a new family of rings, <jats:inline-formula><jats:alternatives><jats:tex-math>${\\mathscr{B}}_{j,k}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>B<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>j<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, whose base field is the finite field <jats:inline-formula><jats:alternatives><jats:tex-math>${\\mathbb {F}}_{p^{r}}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>F<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:msup>\n                        <mml:mrow>\n                          <mml:mi>p<\/mml:mi>\n                        <\/mml:mrow>\n                        <mml:mrow>\n                          <mml:mi>r<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We study the structure of this family of rings and show that each member of the family is a commutative Frobenius ring. We define a Gray map for the new family of rings, study <jats:italic>G<\/jats:italic>-codes, self-dual <jats:italic>G<\/jats:italic>-codes, and reversible <jats:italic>G<\/jats:italic>-codes over this family. In particular, we show that the projection of a <jats:italic>G<\/jats:italic>-code over <jats:inline-formula><jats:alternatives><jats:tex-math>${\\mathscr{B}}_{j,k}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>B<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>j<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>k<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to a code over <jats:inline-formula><jats:alternatives><jats:tex-math>${\\mathscr{B}}_{l,m}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mrow>\n                      <mml:mi>B<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:mi>l<\/mml:mi>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>m<\/mml:mi>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is also a <jats:italic>G<\/jats:italic>-code and the image under the Gray map of a self-dual <jats:italic>G<\/jats:italic>-code is also a self-dual <jats:italic>G<\/jats:italic>-code when the characteristic of the base field is 2. Moreover, we show that the image of a reversible <jats:italic>G<\/jats:italic>-code under the Gray map is also a reversible <jats:inline-formula><jats:alternatives><jats:tex-math>$G^{2^{j+k}}$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>G<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mrow>\n                      <mml:msup>\n                        <mml:mrow>\n                          <mml:mn>2<\/mml:mn>\n                        <\/mml:mrow>\n                        <mml:mrow>\n                          <mml:mi>j<\/mml:mi>\n                          <mml:mo>+<\/mml:mo>\n                          <mml:mi>k<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:msup>\n                    <\/mml:mrow>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-code. The Gray images of these codes are shown to have a rich automorphism group which arises from the algebraic structure of the rings and the groups. Finally, we show that quasi-<jats:italic>G<\/jats:italic> codes, which are the images of <jats:italic>G<\/jats:italic>-codes under the Gray map, are also <jats:italic>G<\/jats:italic><jats:sup><jats:italic>s<\/jats:italic><\/jats:sup>-codes for some <jats:italic>s<\/jats:italic>.<\/jats:p>","DOI":"10.1007\/s12095-021-00487-x","type":"journal-article","created":{"date-parts":[[2021,5,3]],"date-time":"2021-05-03T10:02:58Z","timestamp":1620036178000},"page":"601-616","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["G-codes, self-dual G-codes and reversible G-codes over the ring ${\\mathscr{B}}_{j,k}$"],"prefix":"10.1007","volume":"13","author":[{"given":"S. T.","family":"Dougherty","sequence":"first","affiliation":[]},{"given":"Joe","family":"Gildea","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5206-6480","authenticated-orcid":false,"given":"Adrian","family":"Korban","sequence":"additional","affiliation":[]},{"given":"Serap","family":"\u015eahinkaya","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,5,3]]},"reference":[{"key":"487_CR1","unstructured":"Cengellenmis, Y., Dertli, A., Dougherty, S.T., Korban, A., Sahinkaya, S.: Reversible DNA Codes (in submission)"},{"key":"487_CR2","doi-asserted-by":"publisher","first-page":"559","DOI":"10.1007\/s10623-012-9787-y","volume":"72","author":"Y Cengellenmis","year":"2014","unstructured":"Cengellenmis, Y., Dertli, A., Dougherty, S. T.: Codes over an infinite family of rings with a Gray map. 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