Finite Fields and Their Applications最新文献

{"title":"A spherical extension theorem and applications in positive characteristic","authors":"Doowon Koh ,&nbsp;Thang Pham","doi":"10.1016/j.ffa.2024.102515","DOIUrl":"10.1016/j.ffa.2024.102515","url":null,"abstract":"<div><div>In this paper, we prove an extension theorem for spheres of square radii in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span>, which improves a result obtained by Iosevich and Koh <span><span>[9]</span></span> (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem due to the authors and Lee (2022). Applications on the distance problems will be also discussed.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102515"},"PeriodicalIF":1.2,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427111","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":"Quadratic residue patterns, algebraic curves and a K3 surface","authors":"Valentina Kiritchenko ,&nbsp;Michael Tsfasman ,&nbsp;Serge Vlăduţ ,&nbsp;Ilya Zakharevich","doi":"10.1016/j.ffa.2024.102517","DOIUrl":"10.1016/j.ffa.2024.102517","url":null,"abstract":"<div><div>Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive <em>ℓ</em>-tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102517"},"PeriodicalIF":1.2,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427112","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 the moments subset sum problem through systems of diagonal equations over finite fields","authors":"Juan Francisco Gottig ,&nbsp;Mariana Pérez ,&nbsp;Melina Privitelli","doi":"10.1016/j.ffa.2024.102511","DOIUrl":"10.1016/j.ffa.2024.102511","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span> be the finite field of <em>q</em> elements. Let <em>m</em> and <em>k</em> be positive integers, <span><math><mi>b</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> and <span><math><mi>D</mi><mo>⊂</mo><msub><mrow><mi>F</mi></mrow><mrow><mspace></mspace><mi>q</mi></mrow></msub></math></span> with <span><math><mi>k</mi><mo>≤</mo><mo>|</mo><mi>D</mi><mo>|</mo></math></span>. Our aim is to determine the existence of a subset <span><math><mi>S</mi><mo>⊂</mo><mi>D</mi></math></span> of cardinality <em>k</em> such that <span><math><msub><mrow><mo>∑</mo></mrow><mrow><mi>a</mi><mo>∈</mo><mi>S</mi></mrow></msub><msup><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msup><mo>=</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi></math></span>. This problem is known as the <em>moments subset sum problem</em>. We take a novel approach to this problem by establishing a connection between the existence of such a subset <em>S</em> with the problem of determining if a certain system of diagonal equations has at least one rational solution with distinct coordinates. To achieve this, we study some relevant geometric properties of the set of solutions of the mentioned system. This analysis allows us to provide estimates on the number of rational solutions of systems of diagonal equations and we subsequently apply these results to address the <em>moments subset sum problem</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102511"},"PeriodicalIF":1.2,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427901","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 hyperelliptic Jacobians with complex multiplication by Q(−2+2)","authors":"Tomasz Jędrzejak","doi":"10.1016/j.ffa.2024.102512","DOIUrl":"10.1016/j.ffa.2024.102512","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Consider a one-parameter family of hyperelliptic curves &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;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;y&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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;mn&gt;6&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&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;mn&gt;3&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; defined over &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and their Jacobians &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; where without loss of generality &lt;em&gt;a&lt;/em&gt; is a non-zero squarefree integer. Clearly, the curve &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;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; is a quadratic twist by &lt;em&gt;a&lt;/em&gt; of &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;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Note that &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; has complex multiplication by the quartic field &lt;span&gt;&lt;math&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. For a prime &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;∤&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; we obtain types of reduction (supersingular, superspecial, ordinary) of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; at &lt;em&gt;p&lt;/em&gt; in terms of congruences modulo 16 and the exact formulas for the zeta function of &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;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; for &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;≢&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;. We deduce as conclusions the complete characterization of torsion subgroups of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;, namely &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;tors&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;≃&lt;/mo&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and some information about &lt;span&gt;&lt;math&gt;&lt;mi&gt;rank&lt;/mi&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;J&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;Q&lt;/mi&gt;&lt;m","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102512"},"PeriodicalIF":1.2,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427820","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":"Erasure list-decodable codes and Turán hypercube problems","authors":"Noga Alon","doi":"10.1016/j.ffa.2024.102513","DOIUrl":"10.1016/j.ffa.2024.102513","url":null,"abstract":"<div><div>We observe that several vertex Turán type problems for the hypercube that received a considerable amount of attention in the combinatorial community are equivalent to questions about erasure list-decodable codes. Analyzing a recent construction of Ellis, Ivan and Leader, and determining the Turán density of certain hypergraph augmentations we obtain improved bounds for some of these problems.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102513"},"PeriodicalIF":1.2,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142427819","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 with few weights over finite fields","authors":"Yan Wang ,&nbsp;Jiayi Fan ,&nbsp;Nian Li ,&nbsp;Fangyuan Liu","doi":"10.1016/j.ffa.2024.102509","DOIUrl":"10.1016/j.ffa.2024.102509","url":null,"abstract":"<div><div>Linear codes with a few weights have wide applications in digital signatures, authentication codes, secret sharing protocols and some other fields. Using definition sets to construct linear codes is an effective method. In this paper, we investigate a new defining set and obtain linear codes with four weights, five weights and six weights over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> is an odd prime number. The parameters and weight distribution of the constructed linear code are completely determined by accurately calculating the exponential sum over the finite field.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102509"},"PeriodicalIF":1.2,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142328070","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 explicit values of the UBCT, the LBCT and the DBCT of the inverse function","authors":"Yuying Man ,&nbsp;Nian Li ,&nbsp;Zhen Liu ,&nbsp;Xiangyong Zeng","doi":"10.1016/j.ffa.2024.102508","DOIUrl":"10.1016/j.ffa.2024.102508","url":null,"abstract":"<div><p>Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Upper Boomerang Connectivity Table (UBCT), the Lower Boomerang Connectivity Table (LBCT) and the Double Boomerang Connectivity Table (DBCT) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, there are currently no research results on determining these tables of a function. The inverse function is crucial for constructing S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. Therefore, extensive research has been conducted on the inverse function, exploring various properties related to standard attacks. Thanks to the recent advances in boomerang cryptanalysis, particularly the introduction of concepts such as UBCT, LBCT, and DBCT, this paper aims to further investigate the properties of the inverse function <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>2</mn></mrow></msup></math></span> over <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> for arbitrary <em>n</em>. As a consequence, by carrying out certain finer manipulations of solving specific equations over <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>, we give all entries of the UBCT, LBCT of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <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> for arbitrary <em>n</em>. Besides, based on the results of the UBCT and LBCT for the inverse function, we determine that <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is hard when <em>n</em> is odd. Furthermore, we completely compute all entries of the DBCT of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <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> for arbitrary <em>n</em>. Additionally, we provide the precise number of elements with a given entry by means of the values of some Kloosterman sums. Further, we determine the double boomerang uniformity of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> over <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> for arbitrary <em>n</em>. Our in-depth analysis of the DBCT of <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> contributes to a better evaluation of the S-box's resistance against boomerang attacks.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102508"},"PeriodicalIF":1.2,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240735","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":"Oriented supersingular elliptic curves and Eichler orders of prime level","authors":"Guanju Xiao ,&nbsp;Zijian Zhou ,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2024.102501","DOIUrl":"10.1016/j.ffa.2024.102501","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; be a prime and &lt;em&gt;E&lt;/em&gt; be a supersingular elliptic curve defined over &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;F&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&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;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;em&gt;c&lt;/em&gt; be a prime with &lt;span&gt;&lt;math&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;em&gt;G&lt;/em&gt; be a subgroup of &lt;span&gt;&lt;math&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of order &lt;em&gt;c&lt;/em&gt;. The pair &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is called a supersingular elliptic curve with level-&lt;em&gt;c&lt;/em&gt; structure, and the endomorphism ring &lt;span&gt;&lt;math&gt;&lt;mtext&gt;End&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to an Eichler order with level &lt;em&gt;c&lt;/em&gt;. We construct two kinds of Eichler orders &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; with level &lt;em&gt;c&lt;/em&gt;. Interestingly, we prove that each &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; can represent a primitive reduced binary quadratic form with discriminant &lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;16&lt;/mn&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; respectively. If a curve &lt;em&gt;E&lt;/em&gt; is &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-oriented or &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-oriented, then we prove that &lt;span&gt;&lt;math&gt;&lt;mtext&gt;End&lt;/mtext&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;E&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;G&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is isomorphic to &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; or &lt;span&gt;&lt;math&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;′&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; respectively. Due to the fact that &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-oriented isogenies between &lt;span&gt;&lt;math&gt;&lt;mi&gt;Z&lt;/mi&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;msqrt&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msqrt&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"100 ","pages":"Article 102501"},"PeriodicalIF":1.2,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142240734","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 duality of cyclic codes of length ps over Fpm[u]〈u3〉","authors":"Ahmad Erfanian ,&nbsp;Roghaye Mohammadi Hesari","doi":"10.1016/j.ffa.2024.102500","DOIUrl":"10.1016/j.ffa.2024.102500","url":null,"abstract":"<div><p>In this paper, we determine the dual codes of cyclic codes of length <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mfrac><mrow><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub><mo>[</mo><mi>u</mi><mo>]</mo></mrow><mrow><mo>〈</mo><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>〉</mo></mrow></mfrac></math></span>, where <em>p</em> is a prime number and <span><math><msup><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mn>0</mn></math></span>. Also, we improve and give correction of the results stated by B. Kim and J. Lee (2020) in <span><span>[11]</span></span>. Finally, we provide some examples of optimal and near-MDS cyclic codes of length <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> and compute dual of them.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102500"},"PeriodicalIF":1.2,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142136897","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":"Denniston partial difference sets exist in the odd prime case","authors":"James A. Davis ,&nbsp;Sophie Huczynska ,&nbsp;Laura Johnson ,&nbsp;John Polhill","doi":"10.1016/j.ffa.2024.102499","DOIUrl":"10.1016/j.ffa.2024.102499","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Denniston constructed partial difference sets (PDSs) with the parameters &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;/math&gt;&lt;/span&gt; in elementary abelian groups of order &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; for all &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;&lt;&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"99 ","pages":"Article 102499"},"PeriodicalIF":1.2,"publicationDate":"2024-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001382/pdfft?md5=03b1e738d3c4bc750b4b0f4af02289e1&pid=1-s2.0-S1071579724001382-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142129238","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}