{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,2]],"date-time":"2025-10-02T00:44:14Z","timestamp":1759365854179,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T00:00:00Z","timestamp":1759276800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":["www.mdpi.com"],"crossmark-restriction":true},"short-container-title":["Entropy"],"abstract":"<jats:p>We develop algorithms for computing permutation polynomials (PPs) using normalization, so-called F-maps and G-maps, and the Hermite criterion. This allows for a more efficient computation of PPs for larger degrees and for larger finite fields. We use this to improve some lower bounds for M(n,D), the maximum number of permutations on n symbols with a pairwise Hamming distance of D.<\/jats:p>","DOI":"10.3390\/e27101031","type":"journal-article","created":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T11:06:02Z","timestamp":1759316762000},"page":"1031","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient Algorithms for Permutation Arrays from Permutation Polynomials"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2866-6766","authenticated-orcid":false,"given":"Sergey","family":"Bereg","sequence":"first","affiliation":[{"name":"Department of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Brian","family":"Malouf","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Linda","family":"Morales","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7192-9891","authenticated-orcid":false,"given":"Ivan Hal","family":"Sudborough","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,10,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1080\/00029890.1988.11971991","article-title":"When does a polynomial over a finite field permute the elements of the fields?","volume":"95","author":"Lidl","year":"1988","journal-title":"Am. Math. Mon."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"71","DOI":"10.2307\/2324822","article-title":"When does a polynomial over a finite field permute the elements of the fields? II","volume":"100","author":"Lidl","year":"1993","journal-title":"Am. Math. Mon."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"65","DOI":"10.2307\/1967217","article-title":"The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, part II","volume":"11","author":"Dickson","year":"1896\u20131897","journal-title":"Ann. Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1016\/j.ffa.2014.10.001","article-title":"Permutation polynomials over finite fields\u2014A survey of recent advances","volume":"32","author":"Hou","year":"2015","journal-title":"Finite Fields Appl."},{"key":"ref_5","first-page":"1289","article-title":"Constructions for permutation codes in powerline communications","volume":"50","author":"Chu","year":"2004","journal-title":"Des. Codes Cryptogr."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1016\/j.ffa.2012.12.003","article-title":"Permutation polynomials and orthomorphism polynomials of degree six","volume":"20","author":"Shallue","year":"2013","journal-title":"Finite Fields Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1016\/j.ffa.2010.07.001","article-title":"Permutation polynomials of degree 6 or 7 over finite fields of characteristic 2","volume":"16","author":"Li","year":"2010","journal-title":"Finite Fields Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4019","DOI":"10.1109\/TIT.2019.2957354","article-title":"New Lower Bounds for Permutation Codes Using Linear Block Codes","volume":"66","author":"Micheli","year":"2020","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1007\/s10623-016-0321-5","article-title":"New bounds of permutation codes under Hamming metric and Kendall\u2019s \u03c4-metric","volume":"85","author":"Wang","year":"2017","journal-title":"Des. Codes Cryptogr."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1095","DOI":"10.1007\/s10623-017-0381-1","article-title":"Constructing permutation arrays from groups","volume":"86","author":"Bereg","year":"2018","journal-title":"Des. Codes Cryptogr."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1659","DOI":"10.1007\/s10623-022-01039-x","article-title":"Using permutation rational functions to obtain permutation arrays with large Hamming distance","volume":"90","author":"Bereg","year":"2022","journal-title":"Des. Codes Cryptogr."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2105","DOI":"10.1007\/s10623-019-00607-y","article-title":"New lower bounds for permutation arrays using contraction","volume":"87","author":"Bereg","year":"2019","journal-title":"Des. Codes Cryptogr."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"311","DOI":"10.1007\/s10623-019-00684-z","article-title":"Constructing permutation arrays using partition and extension","volume":"88","author":"Bereg","year":"2020","journal-title":"Des. Codes Cryptogr."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1007\/s10623-016-0263-y","article-title":"Extending permutation arrays: Improving MOLS bounds","volume":"83","author":"Bereg","year":"2017","journal-title":"Des. Codes Cryptogr."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1109\/TIT.2004.828150","article-title":"Permutation arrays for powerline communication and mutually orthogonal Latin squares","volume":"50","author":"Colbourn","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3059","DOI":"10.1109\/TIT.2013.2237945","article-title":"An Improvement on the Gilbert-Varshamov bound for permutation codes","volume":"59","author":"Gao","year":"2013","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/s10623-014-9930-z","article-title":"Permutation codes invariant under isometries","volume":"75","author":"Janiszczak","year":"2015","journal-title":"Des. Codes Cryptogr."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1002\/jcd.21661","article-title":"Isometry invariant permutation codes and mutually orthogonal Latin squares","volume":"27","author":"Janiszczak","year":"2019","journal-title":"J. Combin. Des."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1007\/s10623-011-9551-8","article-title":"A new table of permutation codes","volume":"63","author":"Smith","year":"2012","journal-title":"Des. Codes Cryptogr."},{"key":"ref_20","unstructured":"Bereg, S., Malouf, B., Morales, L., Stanley, T., Sudborough, I.H., and Wong, A. (2020). Equivalence relations for computing permutation polynomials. arXiv."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ffa.2019.05.001","article-title":"A classification of permutation polynomials of degree 7 over finite fields","volume":"59","author":"Fan","year":"2019","journal-title":"Finite Fields Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1017\/S0004972719000674","article-title":"Permutation polynomials of degree 8 over finite fields of odd characteristic","volume":"101","author":"Fan","year":"2019","journal-title":"Bull. Aust. Math. Soc."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.ffa.2020.101662","article-title":"Permutation polynomials of degree 8 over finite fields of characteristic 2","volume":"64","author":"Fan","year":"2020","journal-title":"Finite Fields Appl."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Lidl, R., and Niederreiter, H. (1994). Introduction to Finite Fields and Their Applications, Cambridge University Press. [Revised ed.].","DOI":"10.1017\/CBO9781139172769"},{"key":"ref_25","first-page":"750","article-title":"Sur les fonctions de sept lettres","volume":"57","author":"Hermite","year":"1854","journal-title":"C. R. Acad. Sci. Paris"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2335","DOI":"10.1007\/s10623-019-00621-0","article-title":"A note on good permutation codes from Reed\u2013Solomon codes","volume":"87","author":"Sobhani","year":"2019","journal-title":"Des. Codes Cryptogr."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/10\/1031\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,1]],"date-time":"2025-10-01T11:13:09Z","timestamp":1759317189000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/10\/1031"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,1]]},"references-count":26,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["e27101031"],"URL":"https:\/\/doi.org\/10.3390\/e27101031","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,1]]}}}