Finite Fields and Their Applications最新文献

{"title":"Several classes of permutation polynomials based on the AGW criterion over the finite field F22m","authors":"Guanghui Li ,&nbsp;Xiwang Cao","doi":"10.1016/j.ffa.2024.102378","DOIUrl":"10.1016/j.ffa.2024.102378","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> denote the finite field with <em>q</em><span> elements. Permutation<span> polynomials and complete permutation polynomials over finite fields have been widely investigated in recent years due to their applications in cryptography, coding theory and combinatorial design. In this paper, several classes of (complete) permutation polynomials with the form </span></span><span><math><msup><mrow><mo>(</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mi>δ</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mo>(</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mo>(</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>δ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are proposed based on the AGW criterion and some techniques in solving equations over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>b</mi><mi>x</mi></math></span>, <span><math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> and <span><math><mi>b</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. We also determine the compositional inverse of these polynomials in some special cases. Besides, Mesnager (2014) <span>[13]</span> proposed a construction of bent functions by finding some triples of permutation polynomials satisfying a particular property named <span><math><mo>(</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span>. With the help of this approach, several classes of bent functions are presented.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657722","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":"Superzeta functions on function fields","authors":"Kajtaz H. Bllaca ,&nbsp;Jawher Khmiri ,&nbsp;Kamel Mazhouda ,&nbsp;Bouchaïb Sodaïgui","doi":"10.1016/j.ffa.2024.102367","DOIUrl":"10.1016/j.ffa.2024.102367","url":null,"abstract":"<div><p>We study the superzeta functions on function fields as constructed by Voros (see <span>[11, Chapter 10, p.91]</span><span><span>) in the case of the classical Riemann zeta function. Furthermore, we study special values of those functions, relate them to the Li coefficients, deduce some interesting summation formulas, and prove some results about the regularized product of the zeros of </span>zeta functions on function fields.</span></p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139659469","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 LCD codes from skew symmetric Toeplitz matrices","authors":"Kaimin Cheng","doi":"10.1016/j.ffa.2024.102380","DOIUrl":"10.1016/j.ffa.2024.102380","url":null,"abstract":"<div><p>A linear code with complementary dual (or an LCD code) is defined to be a linear code which intersects its dual code trivially. Let <em>I</em><span> be an identity matrix and </span><em>T</em><span> be a Toeplitz matrix<span> of the same order over a finite field. A Double Toeplitz code (or a DT code) is a linear code generated by a generator matrix of the form </span></span><span><math><mo>(</mo><mi>I</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span><span>. In 2021, Shi et al. obtained necessary and sufficient conditions for a Double Toeplitz code to be LCD when </span><em>T</em> is symmetric and tridiagonal. In this paper, by using a result on factoring Dickson polynomials over finite fields, we determine when a Double Toeplitz code is LCD for <em>T</em><span> being a skew symmetric and tridiagonal matrix. In addition, using a concatenation, we construct LCD codes with arbitrary minimum distance from DT codes over extension fields, provided the length of which is increased if necessary.</span></p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657678","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":"Hausdorff dimension for sets of continued fractions of formal Laurent series","authors":"Mumtaz Hussain,&nbsp;Nikita Shulga","doi":"10.1016/j.ffa.2024.102377","DOIUrl":"10.1016/j.ffa.2024.102377","url":null,"abstract":"<div><p>We prove the Hausdorff dimension of various limsup sets over the field of formal power series. Typically, the upper bound is easier to establish by considering the natural covering of the underlying set. To establish the lower bound, we identify a suitable set that serves as a subset of several limsup sets by selecting appropriate values for the involved parameters. To be precise, given a fixed integer <em>m</em> and for all integers <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>, let <span><math><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span> be a real number. Define the set<span><span><span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo><mspace></mspace><mover><mrow><mo>=</mo></mrow><mrow><mtext>def</mtext></mrow></mover><mo>{</mo><mi>x</mi><mo>∈</mo><mi>I</mi><mo>:</mo><mspace></mspace><mi>deg</mi><mo>⁡</mo><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi><mo>+</mo><mi>i</mi></mrow></msub><mo>=</mo><mo>⌊</mo><mi>n</mi><msub><mrow><mi>α</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>⌋</mo><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mspace></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn><mspace></mspace><mspace></mspace><mspace></mspace><mspace></mspace><mphantom><mspace></mspace><mover><mrow><mo>=</mo></mrow><mrow><mtext>def</mtext></mrow></mover><mo>{</mo><mi>x</mi><mo>∈</mo><mi>I</mi></mphantom><mtext>for infinitely many</mtext><mspace></mspace><mi>n</mi><mo>∈</mo><mi>N</mi><mo>}</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>N</mi></math></span> are fixed, and the partial quotients <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> are polynomials of strictly positive degree. We then determine the Hausdorff dimension of this set which establishes an optimal lower bound for various sets of interest, including the key results in <span>[6]</span>, <span>[8]</span>, <span>[9]</span>. Some new applications of our theorem include the lower bound of the Hausdorff dimension of the formal power series analogues of the sets considered in <span>[2]</span>, <span>[20]</span>. We also prove their upper bounds to provide the comprehensive Hausdorff dimension analysis of these sets.</p><p>The main ingredient of the proof lies in the introduction of <em>m</em> probability measures consistently distributed over the Cantor-type subset of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mi>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000170/pdfft?md5=9c99cf37af3b2658475a1b77cc04137c&pid=1-s2.0-S1071579724000170-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139657562","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":"A new direction on constructing irreducible polynomials over finite fields","authors":"Kaimin Cheng","doi":"10.1016/j.ffa.2024.102368","DOIUrl":"10.1016/j.ffa.2024.102368","url":null,"abstract":"<div><p>Let <em>q</em> be a power of a prime <em>p</em> and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field of order <em>q</em>. Let <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be any polynomial in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> and <span><math><mi>Φ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mfrac></math></span>. For any positive integer <em>n</em>, denote <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span> to be the <em>n</em>-th iterate of Φ and <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>ϕ</mi></mrow></msub></math></span> to be the denominator of <span><math><msup><mrow><mi>Φ</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></math></span>. We call <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> inversely stable over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> if <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>ϕ</mi></mrow></msub></math></span> are distinct and irreducible over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> for all <em>n</em>. In this paper, we aim to find a class of inversely stable polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Actually, let <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>:</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, it is proved that <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is inversely stable over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> if and only if <span><math><mi>a</mi><mo>=</mo><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>b</mi><mo>≠</mo><mn>0</mn></math></span>; moreover, if <span><math><mi>ϕ</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is inversely stable over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, then <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>ϕ</mi></mrow></msub></math></span> is of degree <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> for any positive integer <em>n</em><span>. Consequently, an infinite family of irreducible polynomials over </span><span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is obtained.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139579630","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":"New constructions of optimal (r,δ)-LRCs via good polynomials","authors":"Yuan Gao,&nbsp;Siman Yang","doi":"10.1016/j.ffa.2024.102362","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102362","url":null,"abstract":"<div><p>Locally repairable codes (LRCs) are a class of erasure codes that are widely used in distributed storage systems, which allow for efficient recovery of data in the case of node failures or data loss. In 2014, Tamo and Barg introduced Reed-Solomon-like (RS-like) Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span><span>-LRCs based on polynomial evaluation<span>. These constructions rely on the existence of so-called good polynomial that is constant on each of some pairwise disjoint subsets of </span></span><span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. In this paper, we extend the aforementioned constructions of RS-like LRCs and propose new constructions of <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs whose code length can be larger. These new <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs are all distance-optimal, namely, they attain an upper bound on the minimum distance that will be established in this paper. This bound is sharper than the Singleton-type bound in some cases owing to the extra conditions, it coincides with the Singleton-type bound for certain cases. Combining our constructions with known explicit good polynomials of special forms, we can get various explicit Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with new parameters, whose code lengths are all larger than that constructed by the RS-like <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs introduced by Tamo and Barg. Note that the code length of classical RS codes and RS-like LRCs are both bounded by the field size. We explicitly construct the Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with length <span><math><mi>n</mi><mo>=</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>+</mo><mi>δ</mi></math></span> for any positive integers <span><math><mi>r</mi><mo>,</mo><mi>δ</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mo>(</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>|</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. We also show the existence of Singleton-optimal <span><math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math></span>-LRCs with length <span><math><mi>q</mi><mo>+</mo><mi>δ</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>a</mi></mrow></msup></mrow></msub></math></span> (<span><math><mi>a</mi><mo>≥</mo><mn>3</mn></math></span>) provided <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>|</mo><mo>(</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>, <span><math><mo>(</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>|</mo><m","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139548935","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":"A note on t-designs in isodual codes","authors":"Madoka Awada ,&nbsp;Tsuyoshi Miezaki ,&nbsp;Akihiro Munemasa ,&nbsp;Hiroyuki Nakasora","doi":"10.1016/j.ffa.2024.102366","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102366","url":null,"abstract":"<div><p>In the present paper, we construct 3-designs using extended binary quadratic residue codes and their dual codes.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139548937","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":"Permutation rational functions over quadratic extensions of finite fields","authors":"Ruikai Chen ,&nbsp;Sihem Mesnager","doi":"10.1016/j.ffa.2024.102365","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102365","url":null,"abstract":"<div><p><span>Permutation rational functions over finite fields have attracted much attention in recent years. In this paper, we introduce a class of permutation rational functions over </span><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><span>, whose numerators are so-called </span><em>q</em>-quadratic polynomials. To this end, we will first determine the exact number of zeros of a special <em>q</em>-quadratic polynomial in <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><span>, by calculating character sums related to quadratic forms of </span><span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>/</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Then given some rational function, we can demonstrate whether it induces a permutation of <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>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139548943","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 over Fq3 from bijections of PG(2,q)","authors":"Longjiang Qu ,&nbsp;Kangquan Li","doi":"10.1016/j.ffa.2024.102364","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102364","url":null,"abstract":"<div><p><span>Over the past several years, there are numerous papers about permutation polynomials of the form </span><span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>h</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> over <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><span>. A bijection between the multiplicative subgroup </span><span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> of <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> and the projective line <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> plays a very important role in the research. In this paper, we mainly construct permutation polynomials of the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>h</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span><span> from bijections of the projective plane </span><span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. A bijection from the multiplicative subgroup <span><math><msub><mrow><mi>μ</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span> to <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> is studied, which is a key theorem of this paper. On this basis, some explicit permutation polynomials of the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>r</mi></mrow></msup><mi>h</mi><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span> are constructed from the collineation of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, <em>d</em><span>-homogeneous monomials, 2-homogeneous permutations. It is worth noting that although the bijections of </span><span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> are simple, the corresponding permutation polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139548936","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":"Matrices in M2[Fq[T]] with quadratic minimal polynomial","authors":"Jacobus Visser van Zyl","doi":"10.1016/j.ffa.2024.102361","DOIUrl":"10.1016/j.ffa.2024.102361","url":null,"abstract":"<div><p>By a result of Latimer and MacDuffee, there are a finite number of equivalence classes of <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span> with minimum polynomial <span><math><mi>p</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span>, where <em>p</em> is an <span><math><msup><mrow><mi>n</mi></mrow><mrow><mtext>th</mtext></mrow></msup></math></span> degree polynomial, irreducible over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span>. In this paper, we develop an algorithm for finding a canonical representative of each matrix class, for <span><math><mi>p</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mi>Γ</mi><mi>X</mi><mo>−</mo><mi>Δ</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></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000017/pdfft?md5=c8dd8335361741fc6c65ac182f4475aa&pid=1-s2.0-S1071579724000017-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139475554","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}