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Codes Cryptogr."],"published-print":{"date-parts":[[2025,10]]},"abstract":"Abstract<\/jats:title>\n The \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-additive codes are subgroups of \n \n $$\\mathbb {Z}_2^{\\alpha _1} \\times \\mathbb {Z}_4^{\\alpha _2} \\times \\mathbb {Z}_8^{\\alpha _3}$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n <\/mml:msubsup>\n \u00d7<\/mml:mo>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n \n \u03b1<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:msubsup>\n \u00d7<\/mml:mo>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n \n \u03b1<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n <\/mml:msubsup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. A \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard code is a Hadamard code which is the Gray map image of a \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-additive code. A recursive construction of \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-additive Hadamard codes of type \n \n $$(\\alpha _1,\\alpha _2, \\alpha _3;t_1,t_2, t_3)$$<\/jats:tex-math>\n \n \n (<\/mml:mo>\n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u03b1<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u03b1<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n \u037e<\/mml:mo>\n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with \n \n $$\\alpha _1 \\ne 0$$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$\\alpha _2 \\ne 0$$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$\\alpha _3 \\ne 0$$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$t_1\\ge 1$$<\/jats:tex-math>\n \n \n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$t_2 \\ge 0$$<\/jats:tex-math>\n \n \n \n t<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and \n \n $$t_3\\ge 1$$<\/jats:tex-math>\n \n \n \n t<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is known. In this paper, we generalize some known results for \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard codes to \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard codes with \n \n $$\\alpha _1 \\ne 0$$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, \n \n $$\\alpha _2 \\ne 0$$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and \n \n $$\\alpha _3 \\ne 0$$<\/jats:tex-math>\n \n \n \n \u03b1<\/mml:mi>\n 3<\/mml:mn>\n <\/mml:msub>\n \u2260<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. First, we show for which types the corresponding \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard codes of length \n \n $$2^t$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n t<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for \n \n $$3 \\le t \\le 11$$<\/jats:tex-math>\n \n \n 3<\/mml:mn>\n \u2264<\/mml:mo>\n t<\/mml:mi>\n \u2264<\/mml:mo>\n 11<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard codes, which are not equivalent to any other constructed \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 8<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard code, nor to any \n \n $$\\mathbb {Z}_2\\mathbb {Z}_4$$<\/jats:tex-math>\n \n \n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard code, nor to any previously constructed \n \n $$\\mathbb {Z}_{2^s}$$<\/jats:tex-math>\n \n \n Z<\/mml:mi>\n \n 2<\/mml:mn>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>-linear Hadamard code with \n \n $$s\\ge 2$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, with the same length \n \n $$2^t$$<\/jats:tex-math>\n \n \n 2<\/mml:mn>\n t<\/mml:mi>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s10623-025-01696-8","type":"journal-article","created":{"date-parts":[[2025,7,20]],"date-time":"2025-07-20T07:18:07Z","timestamp":1752995887000},"page":"4567-4594","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linearity and classification of $$\\mathbb {Z}_2\\mathbb {Z}_4\\mathbb {Z}_8$$-linear Hadamard codes"],"prefix":"10.1007","volume":"93","author":[{"given":"Dipak K.","family":"Bhunia","sequence":"first","affiliation":[]},{"given":"Cristina","family":"Fern\u00e1ndez-C\u00f3rdoba","sequence":"additional","affiliation":[]},{"given":"Merc\u00e8","family":"Villanueva","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,7,20]]},"reference":[{"key":"1696_CR1","volume-title":"Designs and Their Codes","author":"EF Assmus","year":"1994","unstructured":"Assmus E.F., Key J.D.: Designs and Their Codes. 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