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Comput."],"published-print":{"date-parts":[[2023,8]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {C}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>C<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> as a module over a principal ideal domain. Hence, <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {C}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>C<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> has a generator polynomial matrix (GPM) <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textbf{G}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>G<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> that satisfies an identical equation. We prove a GPM formula for the Euclidean dual <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {C}^\\perp $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mrow>\n                      <mml:mi>C<\/mml:mi>\n                    <\/mml:mrow>\n                    <mml:mo>\u22a5<\/mml:mo>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> using the identical equation of the Hermite normal form of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\textbf{G}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>G<\/mml:mi>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code.\n<\/jats:p>","DOI":"10.1007\/s12190-023-01889-1","type":"journal-article","created":{"date-parts":[[2023,6,28]],"date-time":"2023-06-28T04:02:27Z","timestamp":1687924947000},"page":"3459-3487","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Two-sided Galois duals of multi-twisted codes"],"prefix":"10.1007","volume":"69","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4184-9222","authenticated-orcid":false,"given":"Ramy","family":"Taki\u00a0Eldin","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,6,28]]},"reference":[{"key":"1889_CR1","doi-asserted-by":"publisher","first-page":"96","DOI":"10.1016\/j.ffa.2016.12.002","volume":"45","author":"N Aydin","year":"2017","unstructured":"Aydin, N., Halilovi\u0107, A.: A generalization of quasi-twisted codes: multi-twisted codes. 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