{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T22:16:01Z","timestamp":1775081761559,"version":"3.50.1"},"reference-count":10,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T00:00:00Z","timestamp":1647907200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T00:00:00Z","timestamp":1647907200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Universiti Sains Malaysia (USM) Research University (RU) Grant","award":["no.1001\/PMATHS\/8011037"],"award-info":[{"award-number":["no.1001\/PMATHS\/8011037"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2022,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Zero-divisor codes are codes constructed using group rings where their generators are zero-divisors. Generally, zero-divisor codes can be equivalent despite their associated groups are non-isomorphic, leading to the proposed conjecture \u201cEvery dihedral zero-divisor code has an equivalent form of cyclic zero-divisor code\u201d. This paper is devoted to study equivalence of zero-divisor codes in <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_2G$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mi>G<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> having generators from the 2-nilradical of <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_2G$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mi>G<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, consisting of all nilpotents of nilpotency degree 2 of <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_2G$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mi>G<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Essentially, algebraic structures of 2-nilradicals are first studied in general for both commutative and non-commutative <jats:inline-formula><jats:alternatives><jats:tex-math>$$F_2G$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>F<\/mml:mi>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mi>G<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> before specialized into the case when <jats:italic>G<\/jats:italic> is cyclic and dihedral. Then, results are used to study the conjecture above in the cases where the codes generators are from their respective 2-nilradicals.<\/jats:p>","DOI":"10.1007\/s10623-022-01025-3","type":"journal-article","created":{"date-parts":[[2022,3,22]],"date-time":"2022-03-22T18:23:17Z","timestamp":1647973397000},"page":"1127-1138","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On equivalence of cyclic and dihedral zero-divisor codes having nilpotents of nilpotency degree two as generators"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0591-3111","authenticated-orcid":false,"given":"Kai Lin","family":"Ong","sequence":"first","affiliation":[]},{"given":"Miin Huey","family":"Ang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,22]]},"reference":[{"key":"1025_CR1","first-page":"31","volume":"3","author":"SD Berman","year":"1967","unstructured":"Berman S.D.: On the theory of group codes. 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