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Codes Cryptogr."],"published-print":{"date-parts":[[2023,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we study Butson Hadamard matrices, and codes over finite rings coming from these matrices in logarithmic form, called BH-codes. We introduce a new morphism of Butson Hadamard matrices through a generalized Gray map on the matrices in logarithmic form, which is comparable to the morphism given in a recent note of \u00d3 Cath\u00e1in and Swartz. That is, we show how, if given a Butson Hadamard matrix over the<jats:inline-formula><jats:alternatives><jats:tex-math>$$k{\\mathrm{th}}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>k<\/mml:mi><mml:mi>th<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>roots of unity, we can construct a larger Butson matrix over the<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell \\mathrm{th}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u2113<\/mml:mi><mml:mi>th<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>roots of unity for any<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u2113<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>dividing<jats:italic>k<\/jats:italic>, provided that any prime<jats:italic>p<\/jats:italic>dividing<jats:italic>k<\/jats:italic>also divides<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\ell $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u2113<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We prove that a<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>Z<\/mml:mi><mml:msup><mml:mi>p<\/mml:mi><mml:mi>s<\/mml:mi><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-additive code with<jats:italic>p<\/jats:italic>a prime number is isomorphic as a group to a BH-code over<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>Z<\/mml:mi><mml:msup><mml:mi>p<\/mml:mi><mml:mi>s<\/mml:mi><\/mml:msup><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and the image of this BH-code under the Gray map is a BH-code over<jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathbb {Z}}_p$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msub><mml:mi>Z<\/mml:mi><mml:mi>p<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>(binary Hadamard code 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\"><mml:mrow><mml:mi>p<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>). Further, we investigate the inherent propelinear structure of these codes (and their images) when the Butson matrix is cocyclic. Some structural properties of these codes are studied and examples are provided.<\/jats:p>","DOI":"10.1007\/s10623-022-01110-7","type":"journal-article","created":{"date-parts":[[2022,9,17]],"date-time":"2022-09-17T12:04:21Z","timestamp":1663416261000},"page":"333-351","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Butson full propelinear codes"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5142-8451","authenticated-orcid":false,"given":"Jos\u00e9 Andr\u00e9s","family":"Armario","sequence":"first","affiliation":[]},{"given":"Ivan","family":"Bailera","sequence":"additional","affiliation":[]},{"given":"Ronan","family":"Egan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,17]]},"reference":[{"issue":"4","key":"1110_CR1","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1007\/s10623-020-00827-7","volume":"89","author":"JA Armario","year":"2021","unstructured":"Armario J.A., Bailera I., Egan R.: Generalized Hadamard full propelinear codes. 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