Finite Fields and Their Applications最新文献

{"title":"Asymptotic distributions of the number of zeros of random polynomials in Hayes equivalence class over a finite field","authors":"Zhicheng Gao","doi":"10.1016/j.ffa.2024.102524","DOIUrl":"10.1016/j.ffa.2024.102524","url":null,"abstract":"<div><div>Hayes equivalence is defined on monic polynomials over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> in terms of the prescribed leading coefficients and the residue classes modulo a given monic polynomial <em>Q</em>. We study the distribution of the number of zeros in a random polynomial over finite fields in a given Hayes equivalence class. It is well known that the number of distinct zeros of a random polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is asymptotically Poisson with mean 1. We show that this is also true for random polynomials in any given Hayes equivalence class. Asymptotic formulas are also given for the number of such polynomials when the degree of such polynomials is proportional to <em>q</em> and the degree of <em>Q</em> and the number of prescribed leading coefficients are bounded by <span><math><msqrt><mrow><mi>q</mi></mrow></msqrt></math></span>. When <span><math><mi>Q</mi><mo>=</mo><mn>1</mn></math></span>, the problem is equivalent to the study of the distance distribution in Reed-Solomon codes. Our asymptotic formulas extend some earlier results and imply that all words for a large family of Reed-Solomon codes are ordinary, which further supports the well-known <em>Deep-Hole</em> Conjecture.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142594089","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":"Quasi-polycyclic and skew quasi-polycyclic codes over Fq","authors":"Tushar Bag,&nbsp;Daniel Panario","doi":"10.1016/j.ffa.2024.102536","DOIUrl":"10.1016/j.ffa.2024.102536","url":null,"abstract":"<div><div>In this research, our focus is on investigating 1-generator right quasi-polycyclic (QPC) codes over fields. We provide a detailed description of how linear codes with substantial minimum distances can be constructed from QPC codes. We analyze dual QPC codes under various inner products and use them to construct quantum error-correcting codes. Furthermore, our research includes a dedicated section that delves into the area of skew quasi-polycyclic (SQPC) codes, investigating their properties and the role of generators in their construction. This section expands our study to encompass the intriguing area of SQPC codes, offering insights into the non-commutative version of QPC codes, their characteristics and generator structures. Our work deals with the structural properties of QPC, skew polycyclic and SQPC codes, shedding light on their potential for enhancing the field of coding theory.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142561480","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":"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}