{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T15:23:43Z","timestamp":1763997823681,"version":"3.45.0"},"reference-count":16,"publisher":"Springer Science and Business Media LLC","issue":"12","license":[{"start":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T00:00:00Z","timestamp":1756252800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T00:00:00Z","timestamp":1756252800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001631","name":"University College Dublin","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001631","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Des. Codes Cryptogr."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Let\n                    <jats:italic>F<\/jats:italic>\n                    be any field containing the finite field of order\n                    <jats:italic>q<\/jats:italic>\n                    . A\n                    <jats:italic>q<\/jats:italic>\n                    -polynomial\n                    <jats:italic>L<\/jats:italic>\n                    over\n                    <jats:italic>F<\/jats:italic>\n                    is an element of the polynomial ring\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>x<\/jats:italic>\n                    ] with the property that all powers of\n                    <jats:italic>x<\/jats:italic>\n                    that appear in\n                    <jats:italic>L<\/jats:italic>\n                    with nonzero coefficient have exponent a power of\n                    <jats:italic>q<\/jats:italic>\n                    . It is well known that given any ordinary polynomial\n                    <jats:italic>f<\/jats:italic>\n                    in\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>x<\/jats:italic>\n                    ], there exists a\n                    <jats:italic>q<\/jats:italic>\n                    -polynomial that is divisible by\n                    <jats:italic>f<\/jats:italic>\n                    . We study the smallest degree of such a\n                    <jats:italic>q<\/jats:italic>\n                    -polynomial. This is equivalent to studying the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${{\\,\\mathrm{\\mathbb {F}}\\,}}_q$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mrow>\n                              <mml:mspace\/>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mspace\/>\n                            <\/mml:mrow>\n                            <mml:mi>q<\/mml:mi>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    -span of the roots of\n                    <jats:italic>f<\/jats:italic>\n                    in a splitting field. We relate this quantity to the representation theory of the Galois group of\n                    <jats:italic>f<\/jats:italic>\n                    . As an application we give a simultaneous construction of the binary Golay code of length 24, and the Steiner system on 24 points.\n                  <\/jats:p>","DOI":"10.1007\/s10623-025-01724-7","type":"journal-article","created":{"date-parts":[[2025,8,27]],"date-time":"2025-08-27T12:31:09Z","timestamp":1756297869000},"page":"5159-5177","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Linearization of polynomials in prime characteristic, with applications to the Golay code and Steiner system"],"prefix":"10.1007","volume":"93","author":[{"given":"Rod","family":"Gow","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gary","family":"McGuire","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,8,27]]},"reference":[{"key":"1724_CR1","doi-asserted-by":"crossref","unstructured":"Abhyankar, S.S.: Galois theory on the line in nonzero characteristic. Bull. Amer. Math. Soc. 27, 68\u2013133 (1992).","DOI":"10.1090\/S0273-0979-1992-00270-7"},{"key":"1724_CR2","doi-asserted-by":"crossref","unstructured":"Abhyankar, S.S., Yie I.: Some more Mathieu group coverings in characteristic two. Proc. Amer. Math. Soc 123, 1007\u20131014 (1994).","DOI":"10.1090\/S0002-9939-1994-1239794-1"},{"key":"1724_CR3","doi-asserted-by":"crossref","unstructured":"Assmus, E.F., Mattson H.F.: Perfect codes and the Mathieu groups. Arch. Math. 17, 122\u2013135 (1966).","DOI":"10.1007\/BF01899857"},{"key":"1724_CR4","doi-asserted-by":"crossref","unstructured":"Bary-Soroker, L., Entin A., McKemmie E.: Galois groups of random additive polynomials. Trans. Amer. Math. Soc. 377, 2231\u20132259 (2024).","DOI":"10.1090\/tran\/9098"},{"key":"1724_CR5","doi-asserted-by":"crossref","unstructured":"Berry, N., Dubickas A., Elkies N., Poonen B., Smyth C.: The conjugate dimension of algebraic numbers. Quart. J. Math. 55, 237\u2013252 (2004).","DOI":"10.1093\/qjmath\/55.3.237"},{"key":"1724_CR6","doi-asserted-by":"crossref","unstructured":"Carmichael, R.D.: Tactical configurations of rank two. Amer. J. Math. 53, 217\u2013240 (1931).","DOI":"10.2307\/2370885"},{"key":"1724_CR7","doi-asserted-by":"crossref","unstructured":"Conway, J., McKay J., Trojan A.: Galois groups over function fields of positive characteristic. Proc. Amer. Math. Soc. 138, 1205\u20131212 (2010).","DOI":"10.1090\/S0002-9939-09-10130-2"},{"key":"1724_CR8","unstructured":"Entin, A., Popov, A.: Probabilistic Galois theory in function fields, arXiv 2311.14862"},{"key":"1724_CR9","doi-asserted-by":"crossref","unstructured":"Girstmair, K.: Linear relations between roots of polynomials. Acta Arith. 89, 53\u201396 (1999).","DOI":"10.4064\/aa-89-1-53-96"},{"key":"1724_CR10","doi-asserted-by":"crossref","unstructured":"Goss, D.: Basic structures of function field arithmetic. Springer, New York (1997).","DOI":"10.1007\/978-3-642-61480-4"},{"key":"1724_CR11","doi-asserted-by":"crossref","unstructured":"Ivanov, A.A.: The monster group and Majorana involutions. Cambridge University Press, Cambridge (2009).","DOI":"10.1017\/CBO9780511576812"},{"key":"1724_CR12","unstructured":"Lidl, R., Niederreiter H.: Finite fields. Addison-Wesley, Reading, Massachusetts (1983)."},{"key":"1724_CR13","unstructured":"MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. North-Holland Math. Library. 16 (1977)."},{"key":"1724_CR14","doi-asserted-by":"crossref","unstructured":"Paige, L.J.: A note on the Mathieu groups. Canad. J. Math. 9, 15\u201318 (1957).","DOI":"10.4153\/CJM-1957-003-8"},{"key":"1724_CR15","doi-asserted-by":"crossref","unstructured":"Todd, J.A.: On representations of the Mathieu groups as collineation groups. J. London Math. Soc. 34, 406\u2013416 (1959).","DOI":"10.1112\/jlms\/s1-34.4.406"},{"key":"1724_CR16","doi-asserted-by":"crossref","unstructured":"Todd, J.A.: A representation of the Mathieu group $$M_{24}$$ as a collineation group. Ann. Di. Math. Pure Ed. Appl. 71, 199\u2013238 (1966).","DOI":"10.1007\/BF02413742"}],"container-title":["Designs, Codes and Cryptography"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01724-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10623-025-01724-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10623-025-01724-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,24]],"date-time":"2025-11-24T15:18:25Z","timestamp":1763997505000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10623-025-01724-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,27]]},"references-count":16,"journal-issue":{"issue":"12","published-print":{"date-parts":[[2025,12]]}},"alternative-id":["1724"],"URL":"https:\/\/doi.org\/10.1007\/s10623-025-01724-7","relation":{},"ISSN":["0925-1022","1573-7586"],"issn-type":[{"type":"print","value":"0925-1022"},{"type":"electronic","value":"1573-7586"}],"subject":[],"published":{"date-parts":[[2025,8,27]]},"assertion":[{"value":"4 November 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 November 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"19 August 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 August 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}