Finite Fields and Their Applications最新文献

{"title":"On trivial cyclically covering subspaces of Fqn","authors":"Jing Huang","doi":"10.1016/j.ffa.2024.102423","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102423","url":null,"abstract":"<div><p>A subspace of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is called a cyclically covering subspace if for every vector of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, operating a certain number of cyclic shifts on it, the resulting vector lies in the subspace. In this paper, we study the problem of under what conditions <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> is itself the only covering subspace of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, symbolically, <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, which is an open problem posed in Cameron et al. (2019) <span>[3]</span> and Aaronson et al. (2021) <span>[1]</span>. We apply the primitive idempotents of the cyclic group algebra to attack this problem; when <em>q</em> is relatively prime to <em>n</em>, we obtain a necessary and sufficient condition under which <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>, which completely answers the problem in this case. Our main result reveals that the problem can be fully reduced to that of determining the values of the trace function over finite fields. As consequences, we explicitly determine several infinitely families of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> which satisfy <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140320786","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Constructing permutation polynomials from permutation polynomials of subfields","authors":"Lucas Reis ,&nbsp;Qiang Wang","doi":"10.1016/j.ffa.2024.102415","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102415","url":null,"abstract":"<div><p>In this paper we study the permutational property of polynomials of the form <span><math><mi>f</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>k</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>⋅</mo><mi>M</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>L</mi><mo>,</mo><mi>M</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> are <em>q</em>-linearized polynomials and <span><math><mi>k</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> satisfies a generic condition. We specialize in the case where <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is the linearized <em>q</em>-associate of <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>a</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>−</mo><mi>a</mi><mo>)</mo></math></span>, <em>t</em> is a divisor of <em>n</em> and <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> satisfies <span><math><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi><mo>/</mo><mi>t</mi></mrow></msup><mo>=</mo><mn>1</mn></math></span>. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> from permutations of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub></math></span>, by simply solving a system of independent equations of the form <span><math><msub><mrow><mi>Tr</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub><mo>(</mo><msup><mrow><mi>δ</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>, where the <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>'s are the coefficients of <em>f</em>. In fact, the same method can be ","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140320784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Application of the Cartier operator in coding theory","authors":"Vahid Nourozi","doi":"10.1016/j.ffa.2024.102419","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102419","url":null,"abstract":"<div><p>The <em>a</em>-number is an invariant of the isomorphism class of the <em>p</em>-torsion group scheme. We use the Cartier operator on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow></msup><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msup><mrow><mi>Ω</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>)</mo></math></span> to find a closed formula for the <em>a</em>-number of the form <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mi>v</mi><mo>(</mo><msup><mrow><mi>Y</mi></mrow><mrow><msqrt><mrow><mi>q</mi></mrow></msqrt></mrow></msup><mo>+</mo><mi>Y</mi><mo>−</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>. The application of the computed <em>a</em>-number in coding theory is illustrated by the relationship between the algebraic properties of the curve and the parameters of codes that are supported by it.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140320785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Private information retrieval from locally repairable databases with colluding servers","authors":"Umberto Martínez-Peñas","doi":"10.1016/j.ffa.2024.102421","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102421","url":null,"abstract":"<div><p>We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups <em>g</em>, locality <em>r</em>, local distance <em>δ</em> and dimension <em>k</em>. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only <em>r</em> nodes (out of <span><math><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn></math></span>) are downloaded per server. The PIR scheme achieves the (download) rate <span><math><mi>R</mi><mo>=</mo><mo>(</mo><mi>N</mi><mo>−</mo><mi>k</mi><mo>−</mo><mi>r</mi><mi>t</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>N</mi></math></span>, where <span><math><mi>N</mi><mo>=</mo><mi>g</mi><mi>r</mi></math></span> is the length of the MDS code obtained after removing the local parities, and for any <em>t</em> colluding servers such that <span><math><mi>k</mi><mo>+</mo><mi>r</mi><mi>t</mi><mo>≤</mo><mi>N</mi></math></span>. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000601/pdfft?md5=0fe90fcdc546f6a24d87a8e7912affb8&pid=1-s2.0-S1071579724000601-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140320783","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Primitive elements of finite fields Fqr avoiding affine hyperplanes for q = 4 and q = 5","authors":"Philipp A. Grzywaczyk ,&nbsp;Arne Winterhof","doi":"10.1016/j.ffa.2024.102416","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102416","url":null,"abstract":"<div><p>For a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> with fixed <em>q</em> and <em>r</em> sufficiently large, we prove the existence of a primitive element outside of a set of <em>r</em> many affine hyperplanes for <span><math><mi>q</mi><mo>=</mo><mn>4</mn></math></span> and <span><math><mi>q</mi><mo>=</mo><mn>5</mn></math></span>. This complements earlier results by Fernandes and Reis for <span><math><mi>q</mi><mo>≥</mo><mn>7</mn></math></span>. For <span><math><mi>q</mi><mo>=</mo><mn>3</mn></math></span> the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span> the set consists only of a single element, and such a result is thus not possible.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000558/pdfft?md5=ec9767fa6acb2934aaa8e7ad60735c8c&pid=1-s2.0-S1071579724000558-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140290917","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On permutation quadrinomials from Niho exponents in characteristic two","authors":"Vincenzo Pallozzi Lavorante","doi":"10.1016/j.ffa.2024.102418","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102418","url":null,"abstract":"<div><p>Recently Zheng et al. <span>[18]</span> characterized the coefficients of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> that lead <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> to be a permutation of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> for <span><math><mo>(</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>,</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></math></span>. They left open the question whether those conditions were also necessary. In this paper, we give a positive answer to that question, solving their conjecture.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140295913","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields","authors":"Robert Tichy ,&nbsp;Daniel Windisch","doi":"10.1016/j.ffa.2024.102413","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102413","url":null,"abstract":"<div><p>We study the class of univariate polynomials <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, introduced by Carlitz, with coefficients in the algebraic function field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo></math></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with <em>q</em> elements. It is implicit in the work of Carlitz that these polynomials form an <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo></math></span>-module basis of the ring <span><math><mi>Int</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo><mo>)</mo><mo>=</mo><mo>{</mo><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>[</mo><mi>X</mi><mo>]</mo><mo>|</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo><mo>)</mo><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo><mo>}</mo></math></span> of integer-valued polynomials on the polynomial ring <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo></math></span>. This stands in close analogy to the famous fact that a <span><math><mi>Z</mi></math></span>-module basis of the ring <span><math><mi>Int</mi><mo>(</mo><mi>Z</mi><mo>)</mo></math></span> is given by the binomial polynomials <span><math><mo>(</mo><mtable><mtr><mtd><mi>X</mi></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></math></span>.</p><p>We prove, for <span><math><mi>k</mi><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>, where <em>s</em> is a non-negative integer, that <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is irreducible in <span><math><mi>Int</mi><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>t</mi><mo>]</mo><mo>)</mo></math></span> and that it is even absolutely irreducible, that is, all of its powers <span><math><msubsup><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> with <span><math><mi>m</mi><mo>&gt;</mo><mn>0</mn></math></span> factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that <span><math><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is not even irreducible if <em>k</em> is not a power of <em>q</em>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000522/pdfft?md5=03cc03a1cb4e3126319ead5a88957a4f&pid=1-s2.0-S1071579724000522-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140290863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"New classes of permutation trinomials of F22m","authors":"Akshay Ankush Yadav ,&nbsp;Indivar Gupta ,&nbsp;Harshdeep Singh ,&nbsp;Arvind Yadav","doi":"10.1016/j.ffa.2024.102414","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102414","url":null,"abstract":"<div><p>In recent years, there have been a lot of research towards finding conditions under which the trinomial <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>α</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>β</mi><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><mn>1</mn><mo>)</mo></math></span> permutes <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> with <span><math><mi>α</mi><mo>&gt;</mo><mi>β</mi></math></span> and <em>r</em> being positive integers. The authors of <span>[6]</span>, <span>[10]</span>, <span>[24]</span> have determined these conditions when <span><math><mi>α</mi><mo>≤</mo><mn>5</mn></math></span> for certain values of <em>β</em> and <em>r</em>. In this paper, we work for <span><math><mi>α</mi><mo>=</mo><mn>6</mn></math></span> and determine four new classes of such permutation trinomials. Our contribution encompasses the investigation of these unexplored classes. Additionally, we analyze their quasi-multiplicative equivalence with already known permutation trinomials for <span><math><mi>m</mi><mo>≥</mo><mn>1</mn></math></span>. Through our research, we demonstrate that two of these determined classes are new, and for others, we explicitly compute the exponent for which they become equivalent.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140295912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Plane curves with a large linear automorphism group in characteristic p","authors":"Herivelto Borges ,&nbsp;Gábor Korchmáros ,&nbsp;Pietro Speziali","doi":"10.1016/j.ffa.2024.102402","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102402","url":null,"abstract":"<div><p>Let <em>G</em> be a subgroup of the three dimensional projective group <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of order <em>q</em>, viewed as a subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span> where <em>K</em> is an algebraic closure of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. For <span><math><mi>G</mi><mo>≅</mo><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and for the seven nonsporadic, maximal subgroups <em>G</em> of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, we investigate the (projective, irreducible) plane curves defined over <em>K</em> that are left invariant by <em>G</em>. For each, we compute the minimum degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of <em>G</em>-invariant curves, provide a classification of all <em>G</em>-invariant curves of degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and determine the first gap <span><math><mi>ε</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> in the spectrum of the degrees of all <em>G</em>-invariant curves. We show that the curves of degree <span><math><mi>d</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> belong to a pencil depending on <em>G</em>, unless they are uniquely determined by <em>G</em>. For most examples of plane curves left invariant by a large subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, the whole automorphism group of the curve is linear, i.e., a subgroup of <span><math><mtext>PGL</mtext><mo>(</mo><mn>3</mn><mo>,</mo><mi>K</mi><mo>)</mo></math></span>. Although this appears to be a general behavior, we show that the opposite case can also occur for some irreducible plane curves, that is, the curve has a large group of linear automorphisms, but its full automorphism group is nonlinear.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140190950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a class of permutation polynomials and their inverses","authors":"Ruikai Chen ,&nbsp;Sihem Mesnager","doi":"10.1016/j.ffa.2024.102403","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102403","url":null,"abstract":"<div><p>We introduce a class of permutation polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> that can be written in the form <span><math><mfrac><mrow><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></mfrac></math></span> or <span><math><mfrac><mrow><mi>L</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow><mrow><mi>x</mi></mrow></mfrac></math></span> for some <em>q</em>-linear polynomial <em>L</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. Specifically, we present those permutation polynomials explicitly as well as their inverses. In addition, more permutation polynomials can be derived in a more general form.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140145169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}