Finite Fields and Their Applications最新文献

{"title":"Self-orthogonal cyclic codes with good parameters","authors":"Jiayuan Zhang,&nbsp;Xiaoshan Kai,&nbsp;Ping Li","doi":"10.1016/j.ffa.2024.102534","DOIUrl":"10.1016/j.ffa.2024.102534","url":null,"abstract":"<div><div>The construction of self-orthogonal codes is an interesting topic due to their wide applications in communication and cryptography. In this paper, we construct several families of self-orthogonal cyclic codes with length <span><math><mi>n</mi><mo>=</mo><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>λ</mi></mrow></mfrac></math></span>, where <span><math><mi>λ</mi><mo>|</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span> and <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span> is odd. It is proved that there exist <em>q</em>-ary self-orthogonal cyclic codes with parameters <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>≥</mo><mi>d</mi><mo>]</mo></math></span> for even prime power <em>q</em>, and <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mn>1</mn><mo>,</mo><mo>≥</mo><mi>d</mi><mo>]</mo></math></span> or <span><math><mo>[</mo><mi>n</mi><mo>,</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mo>≥</mo><mi>d</mi><mo>]</mo></math></span> for odd prime power <em>q</em>, where <em>d</em> is significantly better than the square-root bound. These several families of self-orthogonal cyclic codes contain some optimal linear codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102534"},"PeriodicalIF":1.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561482","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":"Linear codes from planar functions and related covering codes","authors":"Yanan Wu,&nbsp;Yanbin Pan","doi":"10.1016/j.ffa.2024.102535","DOIUrl":"10.1016/j.ffa.2024.102535","url":null,"abstract":"<div><div>Linear codes with few weights have wide applications in consumer electronics, data storage system and secret sharing. In this paper, by virtue of planar functions, several infinite families of <em>l</em>-weight linear codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> are constructed, where <em>l</em> can be any positive integer and <em>p</em> is a prime number. The weight distributions of these codes are determined completely by utilizing certain approach on exponential sums. Experiments show that some (almost) optimal codes in small dimensions can be produced from our results. Moreover, the related covering codes are also investigated.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102535"},"PeriodicalIF":1.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561494","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":"Improvements of the Hasse-Weil-Serre bound over global function fields","authors":"Jinjoo Yoo ,&nbsp;Yoonjin Lee","doi":"10.1016/j.ffa.2024.102538","DOIUrl":"10.1016/j.ffa.2024.102538","url":null,"abstract":"<div><div>We improve the Hasse-Weil-Serre bound over a global function field <em>K</em> with relatively large genus in terms of the ramification behavior of the finite places and the infinite places for <span><math><mi>K</mi><mo>/</mo><mi>k</mi></math></span>, where <em>k</em> is the rational 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>. Furthermore, we improve the Hasse-Weil-Serre bound over a global function field <em>K</em> in terms of the defining equation of <em>K</em>. As an application of our main result, we apply our bound to some well-known extensions: <em>Kummer extensions</em> and <em>elementary abelian p-extensions</em>, where <em>p</em> is the characteristic of <em>k</em>. In fact, elementary abelian <em>p</em>-extensions include <em>Artin-Schreier type extensions</em>, <em>Artin-Schreier extensions</em>, and <em>Suzuki function fields</em>. Moreover, we present infinite families of global function fields for Kummer extensions, Artin-Schreier type extensions, and elementary abelian <em>p</em>-extensions but not Artin-Schreier type extensions, which meet our improved bound: our bound is a sharp bound in these families. We also compare our new bound with some known data given in <span><span>manypoints.org</span><svg><path></path></svg></span>, which is the database on the rational points of algebraic curves. This comparison shows a meaningful improvement of our results on the bound of the number of the rational places of <em>K</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102538"},"PeriodicalIF":1.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561483","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 the cyclotomic field Q(e2πi/p) and Zhi-Wei Sun's conjecture on det Mp","authors":"Li-Yuan Wang ,&nbsp;Hai-Liang Wu","doi":"10.1016/j.ffa.2024.102533","DOIUrl":"10.1016/j.ffa.2024.102533","url":null,"abstract":"<div><div>In 2019, Zhi-Wei Sun posed an interesting conjecture on certain determinants with Legendre symbol entries. In this paper, by using the arithmetic properties of <em>p</em>-th cyclotomic field and the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, we confirm this conjecture.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102533"},"PeriodicalIF":1.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561481","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":"Optimal quinary cyclic codes with three zeros","authors":"Jinmei Fan ,&nbsp;Xiangyong Zeng","doi":"10.1016/j.ffa.2024.102537","DOIUrl":"10.1016/j.ffa.2024.102537","url":null,"abstract":"<div><div>Optimal cyclic codes have received a lot of attention and much progress has been made. However, little is known about optimal quinary cyclic codes. In this paper, by analyzing irreducible factors of certain polynomials over finite fields and utilizing multivariate method, three classes of optimal quinary cyclic codes with parameters <span><math><mo>[</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>5</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>2</mn><mi>m</mi><mo>−</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span> and three zeros are presented.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102537"},"PeriodicalIF":1.2,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552332","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 certain maximal curves related to Chebyshev polynomials","authors":"Guilherme Dias ,&nbsp;Saeed Tafazolian ,&nbsp;Jaap Top","doi":"10.1016/j.ffa.2024.102521","DOIUrl":"10.1016/j.ffa.2024.102521","url":null,"abstract":"<div><div>This paper studies curves defined using Chebyshev polynomials <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over finite fields. Given the hyperelliptic curve <span><math><mi>C</mi></math></span> corresponding to the equation <span><math><msup><mrow><mi>v</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo></math></span>, the prime powers <span><math><mi>q</mi><mo>≡</mo><mn>3</mn><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn></math></span> are determined such that <span><math><msub><mrow><mi>φ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is separable and <span><math><mi>C</mi></math></span> is maximal 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>. This extends a result from <span><span>[30]</span></span> that treats the special cases <span><math><mn>2</mn><mo>|</mo><mi>d</mi></math></span> as well as <em>d</em> a prime number. In particular a proof of <span><span>[30, Conjecture 1.7]</span></span> is presented. Moreover, we give a complete description of the pairs <span><math><mo>(</mo><mi>d</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> such that the projective closure of the plane curve defined by <span><math><msup><mrow><mi>v</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>=</mo><msub><mrow><mi>φ</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo></math></span> is smooth and maximal 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>.</div><div>A number of analogous maximality results are discussed.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102521"},"PeriodicalIF":1.2,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534155","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 constructions of permutation polynomials of the form x+γTrqq2(h(x)) over finite fields with even characteristic","authors":"Sha Jiang,&nbsp;Mu Yuan,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2024.102522","DOIUrl":"10.1016/j.ffa.2024.102522","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Permutation polynomials over finite fields are widely used in cryptography, coding theory, and combinatorial design. Particularly, permutation polynomials of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Tr&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; have been studied by many researchers and applied to lift minimal blocking sets. In this paper, we further investigate permutation polynomials of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Tr&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; over finite fields with even characteristic. On the one hand, guided by the idea of choosing functions &lt;em&gt;h&lt;/em&gt; with a low &lt;em&gt;q&lt;/em&gt;-degree, we completely determine the sufficient and necessary conditions of &lt;em&gt;γ&lt;/em&gt; for six classes of polynomials of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Tr&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; (&lt;span&gt;&lt;math&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;…&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;) to be permutations. These results determine the sizes of directions of these six functions, which is generally difficult. On the other hand, we slightly generalize the above idea and construct other six classes of permutation polynomials of the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;γ&lt;/mi&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;Tr&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102522"},"PeriodicalIF":1.2,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534067","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":"An approach to normal polynomials through symmetrization and symmetric reduction","authors":"Darien Connolly ,&nbsp;Calvin George ,&nbsp;Xiang-dong Hou ,&nbsp;Adam Madro ,&nbsp;Vincenzo Pallozzi Lavorante","doi":"10.1016/j.ffa.2024.102525","DOIUrl":"10.1016/j.ffa.2024.102525","url":null,"abstract":"<div><div>An irreducible polynomial <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span> of degree <em>n</em> is <em>normal</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> if and only if its roots <span><math><mi>r</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup></math></span> satisfy the condition <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>)</mo><mo>≠</mo><mn>0</mn></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> is the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> circulant determinant. By finding a suitable <em>symmetrization</em> of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (A multiple of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> which is symmetric in <span><math><msub><mrow><mi>X</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>), we obtain a condition on the coefficients of <em>f</em> that is sufficient for <em>f</em> to be normal. This approach works well for <span><math><mi>n</mi><mo>≤</mo><mn>5</mn></math></span> but encounters computational difficulties when <span><math><mi>n</mi><mo>≥</mo><mn>6</mn></math></span>. In the present paper, we consider irreducible polynomials of the form <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span>. For <span><math><mi>n</mi><mo>=</mo><mn>6</mn></math></span> and 7, by an indirect method, we are able to find simple conditions on <em>a</em> that are sufficient for <em>f</em> to be normal. In a more general context, we also explore the normal polynomials of a finite Galois extension through the irreducible characters of the Galois group.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102525"},"PeriodicalIF":1.2,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534066","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":"The compositional inverses of three classes of permutation polynomials over finite fields","authors":"Danyao Wu ,&nbsp;Pingzhi Yuan ,&nbsp;Huanhuan Guan ,&nbsp;Juan Li","doi":"10.1016/j.ffa.2024.102523","DOIUrl":"10.1016/j.ffa.2024.102523","url":null,"abstract":"<div><div>R. Gupta, P. Gahlyan and R.K. Sharma presented three classes of permutation trinomials 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> in Finite Fields and Their Applications. In this paper, we employ the local method to prove that those polynomials are indeed permutation polynomials and provide their compositional inverses.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102523"},"PeriodicalIF":1.2,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534154","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":"Stable binomials over finite fields","authors":"Arthur Fernandes ,&nbsp;Daniel Panario ,&nbsp;Lucas Reis","doi":"10.1016/j.ffa.2024.102520","DOIUrl":"10.1016/j.ffa.2024.102520","url":null,"abstract":"<div><div>In this paper, we study stable binomials over finite fields, i.e., irreducible binomials <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>−</mo><mi>b</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> such that all their iterates are also irreducible over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We obtain a simple criterion on the stability of binomials based on the forward orbit of 0 under the map <span><math><mi>z</mi><mo>↦</mo><msup><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>−</mo><mi>b</mi></math></span>. In particular, our criterion extends the one obtained by Jones and Boston (2011) for the quadratic case. As applications of our main result, we obtain an explicit 1-parameter family of stable quartics over prime fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> with <span><math><mi>p</mi><mo>≡</mo><mn>5</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>24</mn><mo>)</mo></math></span> and also develop an algorithm to test the stability of binomials over finite fields. Finally, building upon a work of Ostafe and Shparlinski (2010), we employ character sums to bound the complexity of such algorithm.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102520"},"PeriodicalIF":1.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142534153","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}