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Codes Cryptogr."],"published-print":{"date-parts":[[2022,4]]},"abstract":"Abstract<\/jats:title>$${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-additive codes of length n<\/jats:italic> are subgroups of $${\\mathbb {Z}}_{p^s}^n$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n n<\/mml:mi>\n <\/mml:msubsup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, and can be seen as a generalization of linear codes over $${\\mathbb {Z}}_2$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, $${\\mathbb {Z}}_4$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, or $${\\mathbb {Z}}_{2^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n 2<\/mml:mn>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> in general. A $${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-linear generalized Hadamard (GH) code is a GH code over $${\\mathbb {Z}}_p$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> which is the image of a $${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-additive code by a generalized Gray map. In this paper, we generalize some known results for $${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-linear GH codes with $$p=2$$<\/jats:tex-math>\n \n p<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> to any odd prime p<\/jats:italic>. First, we show some results related to the generalized Carlet\u2019s Gray map. Then, by using an iterative construction of $${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-additive GH codes of type $$(n;t_1,\\ldots , t_s)$$<\/jats:tex-math>\n \n (<\/mml:mo>\n n<\/mml:mi>\n \u037e<\/mml:mo>\n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, we show for which types the corresponding $${\\mathbb {Z}}_{p^s}$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n \n p<\/mml:mi>\n s<\/mml:mi>\n <\/mml:msup>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-linear GH codes of length $$p^t$$<\/jats:tex-math>\n \n p<\/mml:mi>\n t<\/mml:mi>\n <\/mml:msup>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are nonlinear over $${\\mathbb {Z}}_p$$<\/jats:tex-math>\n \n Z<\/mml:mi>\n p<\/mml:mi>\n <\/mml:msub>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For these codes, we compute the kernel and its dimension, which allow us to give a partial classification. The obtained results for $$p\\ge 3$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2265<\/mml:mo>\n 3<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are different from the case with $$p=2$$<\/jats:tex-math>\n \n p<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>. Finally, the exact number of non-equivalent such codes is given for an infinite number of values of s<\/jats:italic>, t<\/jats:italic>, and any $$p\\ge 2$$<\/jats:tex-math>\n \n p<\/mml:mi>\n \u2265<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula>; by using also the rank as an invariant in some specific cases.<\/jats:p>","DOI":"10.1007\/s10623-022-01026-2","type":"journal-article","created":{"date-parts":[[2022,3,14]],"date-time":"2022-03-14T12:06:52Z","timestamp":1647259612000},"page":"1037-1058","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":15,"title":["On the linearity and classification of $${\\mathbb {Z}}_{p^s}$$-linear generalized hadamard codes"],"prefix":"10.1007","volume":"90","author":[{"given":"Dipak K.","family":"Bhunia","sequence":"first","affiliation":[]},{"given":"Cristina","family":"Fern\u00e1ndez-C\u00f3rdoba","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6179-0833","authenticated-orcid":false,"given":"Merc\u00e8","family":"Villanueva","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,3,14]]},"reference":[{"key":"1026_CR1","unstructured":"Armario, J.A., Bailera, I., Egan, R.: Butson full propelinear codes. preprint arXiv:2010.06206 (2020)"},{"key":"1026_CR2","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781316529836","volume-title":"Designs and Their Codes","author":"EF Assmus","year":"1992","unstructured":"Assmus E.F., Key J.D.: Designs and Their Codes. 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