Finite Fields and Their Applications最新文献

{"title":"Heffter spaces","authors":"M. Buratti ,&nbsp;A. Pasotti","doi":"10.1016/j.ffa.2024.102464","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102464","url":null,"abstract":"<div><p>The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of <em>r</em> mutually orthogonal Heffter systems for any <em>r</em>. Such a set is equivalent to a resolvable partial linear space of degree <em>r</em> whose parallel classes are Heffter systems: this is a new combinatorial design that we call a <em>Heffter space</em>. We present a series of direct constructions of Heffter spaces with odd block size and arbitrarily large degree <em>r</em> obtained with the crucial use of finite fields. Among the applications we establish, in particular, that if <span><math><mi>q</mi><mo>=</mo><mn>2</mn><mi>k</mi><mi>w</mi><mo>+</mo><mn>1</mn></math></span> is a prime power with <em>kw</em> odd and <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, then there are at least <span><math><mo>⌈</mo><mfrac><mrow><mi>w</mi></mrow><mrow><mn>4</mn><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></mfrac><mo>⌉</mo></math></span> mutually orthogonal <em>k</em>-cycle systems of order <em>q</em>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102464"},"PeriodicalIF":1.2,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593494","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":"Odd moments for the trace of Frobenius and the Sato–Tate conjecture in arithmetic progressions","authors":"Kathrin Bringmann ,&nbsp;Ben Kane ,&nbsp;Sudhir Pujahari","doi":"10.1016/j.ffa.2024.102465","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102465","url":null,"abstract":"<div><p>In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the <span><math><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-th moment to the zeroeth moment as the size of the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></mrow></msub></math></span> goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102465"},"PeriodicalIF":1.2,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141593493","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 some congruences and exponential sums","authors":"Moubariz Z. Garaev ,&nbsp;Igor E. Shparlinski","doi":"10.1016/j.ffa.2024.102451","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102451","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; be a fixed small constant, &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; be the finite field of &lt;em&gt;p&lt;/em&gt; elements for prime &lt;em&gt;p&lt;/em&gt;. We consider additive and multiplicative problems in &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; that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be an arbitrary subset of &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;. If &lt;span&gt;&lt;math&gt;&lt;mi&gt;#&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&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;1&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&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;2&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; or if &lt;span&gt;&lt;math&gt;&lt;mi&gt;#&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&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;mo&gt;/&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&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;mo&gt;/&lt;/mo&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; then all, but &lt;span&gt;&lt;math&gt;&lt;mi&gt;O&lt;/mi&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;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; elements of &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; can be represented in the form &lt;em&gt;hm&lt;/em&gt; with &lt;span&gt;&lt;math&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, where &lt;span&gt;&lt;math&gt;&lt;mi&gt;δ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; depends only on &lt;em&gt;ε&lt;/em&gt;. Furthermore, let &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; be an arbitrary interval of length &lt;em&gt;H&lt;/em&gt; and &lt;em&gt;s&lt;/em&gt; be a fixed positive integer. If&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mi&gt;H&lt;/mi&gt;&lt;mo&gt;&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;17&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;35&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;#&lt;/mi&gt;&lt;mi&gt;M&lt;/mi&gt;&lt;mo&gt;&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;17&lt;/mn&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;mn&gt;35&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;ε&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; then the number &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;T&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;6&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; of solutions to the congruence&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&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;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msubsup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102451"},"PeriodicalIF":1.2,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S107157972400090X/pdfft?md5=73d751bad88083ca796c715f3b4d9bad&pid=1-s2.0-S107157972400090X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485792","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":"Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold","authors":"Mrinmoy Datta ,&nbsp;Subrata Manna","doi":"10.1016/j.ffa.2024.102462","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102462","url":null,"abstract":"<div><p>It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math></span> has at most <span><math><mi>d</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn></math></span> points in common with a threefold of degree <em>d</em> defined 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>. He proved the conjecture for <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>. In this paper, we show that the conjecture is true for <span><math><mi>d</mi><mo>=</mo><mn>3</mn></math></span> and <span><math><mi>q</mi><mo>≥</mo><mn>7</mn></math></span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102462"},"PeriodicalIF":1.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485824","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 further study on the Ness-Helleseth function","authors":"Cheng Lyu,&nbsp;Xiaoqiang Wang,&nbsp;Dabin Zheng","doi":"10.1016/j.ffa.2024.102453","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102453","url":null,"abstract":"&lt;div&gt;&lt;p&gt;Let &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;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be a finite field with &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; elements. Ness and Helleseth in &lt;span&gt;[29]&lt;/span&gt; first studied a class of functions 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;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; with the form &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&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;mi&gt;u&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&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;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msubsup&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;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msubsup&gt;&lt;/math&gt;&lt;/span&gt;, which is called the Ness-Helleseth function. The &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; in &lt;span&gt;[29]&lt;/span&gt; and by Zeng et al. for any odd prime &lt;em&gt;p&lt;/em&gt; in &lt;span&gt;[43]&lt;/span&gt; under the condition &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, we continue to study the Ness-Helleseth functions under the condition that &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;≡&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;mod&lt;/mi&gt;&lt;/mrow&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;≠&lt;/mo&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Firstly, we prove that &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a permutation polynomial with differential uniformity not more than 4 if &lt;span&gt;&lt;math&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;η&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. Moreover, for some more special &lt;em&gt;u&lt;/em&gt;, &lt;em&gt;f&lt;/em&gt; is an involution with differential uniformity at most 3. Secondly, we show that &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a locally-APN function for &lt;span&gt;&lt;math&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo&gt;±&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. In addition, the differential spectrum and boomerang spectrum of &lt;span&gt;&lt;math&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;)&lt;/","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102453"},"PeriodicalIF":1.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482840","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":"Cyclic 2-spreads in V(6,q) and flag-transitive linear spaces","authors":"Cian Jameson,&nbsp;John Sheekey","doi":"10.1016/j.ffa.2024.102463","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102463","url":null,"abstract":"<div><p>In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102463"},"PeriodicalIF":1.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001023/pdfft?md5=cd6ef2ef8226a10487b87f668b7b3d4e&pid=1-s2.0-S1071579724001023-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485873","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":"Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2","authors":"Barbara Gatti ,&nbsp;Gábor Korchmáros","doi":"10.1016/j.ffa.2024.102450","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102450","url":null,"abstract":"<div><p>A (projective, geometrically irreducible, non-singular) curve <span><math><mi>X</mi></math></span> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> is <em>maximal</em> if the number <span><math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span> of its <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>-rational points attains the Hasse-Weil upper bound, that is <span><math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>g</mi><mi>q</mi><mo>+</mo><mn>1</mn></math></span> where <span><math><mi>g</mi></math></span> is the genus of <span><math><mi>X</mi></math></span>. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> where <em>p</em> is the characteristic 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>. Doing so we also determine the <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>-isomorphism classes of such curves and describe their full <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>-automorphism groups.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102450"},"PeriodicalIF":1.2,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000893/pdfft?md5=4d7dc430a08ae7fb7ea1e9c89490e861&pid=1-s2.0-S1071579724000893-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141438233","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 multistep strategy for polynomial system solving over finite fields and a new algebraic attack on the stream cipher Trivium","authors":"Roberto La Scala ,&nbsp;Federico Pintore ,&nbsp;Sharwan K. Tiwari ,&nbsp;Andrea Visconti","doi":"10.1016/j.ffa.2024.102452","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102452","url":null,"abstract":"<div><p>In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive evaluation of a subset of variables stepwise, that is, by incrementing the size of such subset each time that an evaluation leads to a polynomial system which is possibly unfeasible to solve. The decision about which evaluation to extend is based on a preprocessing consisting in computing an incomplete Gröbner basis after the current evaluation, which possibly generates linear polynomials that are used to eliminate further variables. If the number of remaining variables in the system is deemed still too high, the evaluation is extended and the preprocessing is iterated. Otherwise, we solve the system by a complete Gröbner basis computation.</p><p>Having in mind cryptanalytic applications, we present an implementation of this strategy in an algorithm called <span>MultiSolve</span> which is designed for polynomial systems having at most one solution. We prove explicit formulas for its complexity which are based on probability distributions that can be easily estimated by performing the proposed preprocessing on a testset of evaluations for different subsets of variables. We prove that an optimal complexity of <span>MultiSolve</span> is achieved by using a full multistep strategy with a maximum number of steps and in turn the standard guess-and-determine strategy, which essentially is a strategy consisting of a single step, is the worst choice. Finally, we extensively study the behaviour of <span>MultiSolve</span> when performing an algebraic attack on the well-known stream cipher <span>Trivium</span>.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"98 ","pages":"Article 102452"},"PeriodicalIF":1.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141429874","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":"Scattered trinomials of Fq6[X] in even characteristic","authors":"Daniele Bartoli ,&nbsp;Giovanni Longobardi ,&nbsp;Giuseppe Marino ,&nbsp;Marco Timpanella","doi":"10.1016/j.ffa.2024.102449","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102449","url":null,"abstract":"<div><p>In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In <span>[9]</span>, <span>[24]</span>, the authors proved that the trinomial <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>X</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></msup></math></span> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span> is scattered under the assumptions that <em>q</em> is odd and <span><math><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>1</mn></math></span>. They also explicitly observed that this is false when <em>q</em> is even. In this paper, we provide a different set of conditions on <em>c</em> for which this trinomial is scattered in the case of even <em>q</em>. Using tools of algebraic geometry in positive characteristic, we show that when <em>q</em> is even and sufficiently large, there are roughly <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> elements <span><math><mi>c</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup></mrow></msub></math></span> such that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is scattered. Also, we prove that the corresponding MRD-codes and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-linear sets of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span> are not equivalent to the previously known ones.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"97 ","pages":"Article 102449"},"PeriodicalIF":1.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000881/pdfft?md5=8a841f52ac38f09246cba1d4c920247c&pid=1-s2.0-S1071579724000881-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the P-construction of algebraic-geometry codes","authors":"R. Toledano ,&nbsp;M. Vides","doi":"10.1016/j.ffa.2024.102448","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102448","url":null,"abstract":"<div><p>We construct algebraic-geometry codes by using projective systems from projective curves over a finite field and the global sections of invertible sheaves on these curves. We also prove a formula for the Hilbert function of a finite set of points in a projective space in terms of the rank of a matrix constructed with the Veronese embedding and we use it to estimate the minimum distance of the dual codes.</p></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"97 ","pages":"Article 102448"},"PeriodicalIF":1.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264015","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}