Finite Fields and Their Applications最新文献

{"title":"Generalizing blocking semiovals in finite projective planes","authors":"Marilena Crupi ,&nbsp;Antonino Ficarra","doi":"10.1016/j.ffa.2025.102688","DOIUrl":"10.1016/j.ffa.2025.102688","url":null,"abstract":"<div><div>Blocking semiovals and the determination of their (minimum) sizes constitute one of the central research topics in finite projective geometry. In this article we introduce the concept of blocking set with the <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-property in a finite projective plane <span><math><mtext>PG</mtext><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, with <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> a line of <span><math><mtext>PG</mtext><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and <em>q</em> a prime power. This notion greatly generalizes that of blocking semioval. We address the question of determining those integers <em>k</em> for which there exists a blocking set of size <em>k</em> with the <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span>-property. To solve this problem, we build new theory which deeply analyzes the interplay between blocking sets in finite projective and affine planes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102688"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570245","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":"Decoding algorithms in group codes","authors":"C. Martínez,&nbsp;F. Molina,&nbsp;A. Piñera-Nicolás","doi":"10.1016/j.ffa.2025.102692","DOIUrl":"10.1016/j.ffa.2025.102692","url":null,"abstract":"<div><div>Group codes are linear codes that can be identified with (two-sided) ideals of a group algebra <span><math><mi>K</mi><mi>G</mi></math></span>. Assuming that <span><math><mi>K</mi><mi>G</mi></math></span> is semisimple, we use its decomposition as the direct sum of two ideals, one of them the group code, to design two decoding algorithms. The first one generalizes Meggitt's algorithm designed for cyclic codes, while the other one is inspired in the decoding algorithm studied in <span><span>[10]</span></span> and aims to improve it.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102692"},"PeriodicalIF":1.2,"publicationDate":"2025-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570386","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":"Euclidean sets with only one distance modulo a prime ideal","authors":"Hiroshi Nozaki","doi":"10.1016/j.ffa.2025.102690","DOIUrl":"10.1016/j.ffa.2025.102690","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;em&gt;X&lt;/em&gt; be a finite set in the Euclidean space &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;. If the squared distance between any two distinct points in &lt;em&gt;X&lt;/em&gt; is an odd integer, then the cardinality of &lt;em&gt;X&lt;/em&gt; is bounded above by &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, as shown by Rosenfeld (1997) or Smith (1995). They proved that there exists a &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;-point set &lt;em&gt;X&lt;/em&gt; in &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; having only odd integral squared distances if and only if &lt;span&gt;&lt;math&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; is congruent to 0 modulo 4. The distances can be interpreted as an element of the finite field &lt;span&gt;&lt;math&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;. We generalize this result for a local ring &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; as follows. Let &lt;em&gt;K&lt;/em&gt; be an algebraic number field that can be embedded into &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Fix an embedding of &lt;em&gt;K&lt;/em&gt; into &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, and &lt;em&gt;K&lt;/em&gt; is interpreted as a subfield of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;K&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; be the ring of integers of &lt;em&gt;K&lt;/em&gt;, and &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; a prime ideal of &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;K&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;. Let &lt;span&gt;&lt;math&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; be the local ring obtained from the localization &lt;span&gt;&lt;math&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;mo&gt;∖&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, which is interpreted as a subring of &lt;span&gt;&lt;math&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;. If the squared distances of &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&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;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; are in &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&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; and each squared distance is congruent to some constant &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≢&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; modulo &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;A&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;, then &lt;span&gt;&lt;math&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;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, as shown by Nozaki (2023). In this paper, we prove that there exists a set &lt;span&gt;&lt;math&gt;&lt;mi&gt;X&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;mi&gt;d&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; attaining the upper bound &lt;span&gt;&lt;math&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;d&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102690"},"PeriodicalIF":1.2,"publicationDate":"2025-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144534171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Permutation polynomials of finite fields of even characteristic from character sums","authors":"Ruikai Chen ,&nbsp;Sihem Mesnager","doi":"10.1016/j.ffa.2025.102684","DOIUrl":"10.1016/j.ffa.2025.102684","url":null,"abstract":"<div><div>In this paper, we investigate permutation polynomials over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> with <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>, focusing on those in the form <span><math><mi>Tr</mi><mo>(</mo><mi>A</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>+</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, where <span><math><mi>A</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> and <em>L</em> is a 2-linear polynomial over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>. By calculating certain character sums, we characterize these permutation polynomials and provide additional constructions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102684"},"PeriodicalIF":1.2,"publicationDate":"2025-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144513839","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":"Extending a result of Carlitz and McConnel to polynomials which are not permutations","authors":"Bence Csajbók","doi":"10.1016/j.ffa.2025.102683","DOIUrl":"10.1016/j.ffa.2025.102683","url":null,"abstract":"<div><div>Let <em>D</em> denote the set of directions determined by the graph of a polynomial <em>f</em> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>, where <em>q</em> is a power of the prime <em>p</em>. If <em>D</em> is contained in a multiplicative subgroup <em>M</em> of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span>, then by a result of Carlitz and McConnel it follows that <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msup><mo>+</mo><mi>b</mi></math></span> for some <span><math><mi>k</mi><mo>∈</mo><mi>N</mi></math></span>. Of course, if <span><math><mi>D</mi><mo>⊆</mo><mi>M</mi></math></span>, then <span><math><mn>0</mn><mo>∉</mo><mi>D</mi></math></span> and hence <em>f</em> is a permutation. If we assume the weaker condition <span><math><mi>D</mi><mo>⊆</mo><mi>M</mi><mo>∪</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>, then <em>f</em> is not necessarily a permutation, but Sziklai conjectured that <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></msup><mo>+</mo><mi>b</mi></math></span> follows also in this case. When <em>q</em> is odd, and the index of <em>M</em> is even, then a result of Ball, Blokhuis, Brouwer, Storme and Szőnyi combined with a result of Göloğlu and McGuire proves the conjecture. Assume <span><math><mi>deg</mi><mo>⁡</mo><mi>f</mi><mo>≥</mo><mn>1</mn></math></span>. We prove that if the size of <span><math><msup><mrow><mi>D</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>D</mi><mo>=</mo><mo>{</mo><msup><mrow><mi>d</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>d</mi><mo>∈</mo><mi>D</mi><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>,</mo><mspace></mspace><msup><mrow><mi>d</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>D</mi><mo>}</mo></math></span> is less than <span><math><mi>q</mi><mo>−</mo><mi>deg</mi><mo>⁡</mo><mi>f</mi><mo>+</mo><mn>2</mn></math></span>, then <em>f</em> is a permutation of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We use this result to prove the conjecture of Sziklai.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102683"},"PeriodicalIF":1.2,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489462","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":"Some new classes of permutation polynomials and their compositional inverses","authors":"Sartaj Ul Hasan,&nbsp;Ramandeep Kaur","doi":"10.1016/j.ffa.2025.102685","DOIUrl":"10.1016/j.ffa.2025.102685","url":null,"abstract":"<div><div>We focus on permutation polynomials of the form <span><math><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>+</mo><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msubsup><msup><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is the finite field with <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> elements, <em>p</em> is a prime number, <em>m</em> is a positive integer, <span><math><msubsup><mrow><mi>Tr</mi></mrow><mrow><mi>m</mi></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msubsup><mo>(</mo><mo>⋅</mo><mo>)</mo></math></span> is the relative trace function from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span>, <span><math><mi>L</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> is a linearized polynomial 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>, and <span><math><mi>s</mi><mo>&gt;</mo><mn>1</mn></math></span> is a positive integer. More precisely, we present six new classes of permutation polynomials 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> of the aforementioned form: one class over finite fields of even characteristic, three classes over finite fields of odd characteristic, and the remaining two over finite fields of arbitrary characteristic. Furthermore, we show that these classes of permutation polynomials are inequivalent to the known ones of the same form. We also provide explicit expressions for the compositional inverses of each of these classes of permutation polynomials.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102685"},"PeriodicalIF":1.2,"publicationDate":"2025-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144489390","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 Fp2-maximal curves of the form y3 = f(x)","authors":"Guilherme Dias,&nbsp;Saeed Tafazolian","doi":"10.1016/j.ffa.2025.102682","DOIUrl":"10.1016/j.ffa.2025.102682","url":null,"abstract":"<div><div>We investigate the maximality of algebraic curves associated with 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. Specifically, we study the curve <span><math><mi>C</mi></math></span> given by <span><math><msup><mrow><mi>v</mi></mrow><mrow><mn>3</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> and determine all finite fields over which these curves attain the Hasse–Weil upper bound. Our results generalize previous work that focused on hyperelliptic curves. Additionally, we examine other related curves of the form <span><math><msup><mrow><mi>y</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> throughout the paper.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102682"},"PeriodicalIF":1.2,"publicationDate":"2025-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322295","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":"Deep holes of twisted Reed-Solomon codes","authors":"Weijun Fang ,&nbsp;Jingke Xu ,&nbsp;Ruiqi Zhu","doi":"10.1016/j.ffa.2025.102680","DOIUrl":"10.1016/j.ffa.2025.102680","url":null,"abstract":"<div><div>The deep holes of a linear code are the vectors that achieve the maximum error distance (covering radius) to the code. Determining the covering radius and deep holes of linear codes is a fundamental problem in coding theory. In this paper, we investigate the problem of deep holes of twisted Reed-Solomon codes. The covering radius and a standard class of deep holes of twisted Reed-Solomon codes <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>A</mi><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> are obtained for a general evaluation set <span><math><mi>A</mi><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Furthermore, we consider the problem of determining all deep holes of the full-length twisted Reed-Solomon codes <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></math></span>. For even <em>q</em>, by utilizing the polynomial method and Gauss sums over finite fields, we prove that the standard deep holes are all the deep holes of <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> with <span><math><mfrac><mrow><mn>3</mn><mi>q</mi><mo>−</mo><mn>4</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>q</mi><mo>−</mo><mn>4</mn></math></span>. For odd <em>q</em>, we adopt a different method and employ the results on some equations over finite fields to show that there are also no other deep holes of <span><math><msub><mrow><mi>TRS</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mi>θ</mi><mo>)</mo></math></span> with <span><math><mfrac><mrow><mn>3</mn><mi>q</mi><mo>+</mo><mn>3</mn><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>−</mo><mn>7</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>q</mi><mo>−</mo><mn>4</mn></math></span>. In addition, for the boundary cases of <span><math><mi>k</mi><mo>=</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>2</mn></math></span> and <span><math><mi>q</mi><mo>−</mo><mn>1</mn></math></span>, we completely determine their deep holes using results on certain character sums.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102680"},"PeriodicalIF":1.2,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290832","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":"Further study on multivariate technique for permutation polynomials","authors":"Mu Yuan,&nbsp;Kangquan Li,&nbsp;Longjiang Qu","doi":"10.1016/j.ffa.2025.102678","DOIUrl":"10.1016/j.ffa.2025.102678","url":null,"abstract":"<div><div>The research of permutation polynomials has been a hot topic due to their wide applications in various areas. Based on the multivariate technique, this paper proposes six classes of permutation trinomials over finite fields with even characteristics via three different approaches. The first approach is a variant of the so-called <span><math><mi>L</mi></math></span>-method. Meanwhile, the presented results generalize some previous results. The second one employs the connections of these polynomials with Dickson polynomials and symmetric polynomials. The third one is based on a well-known lemma and some arithmetics over the multiplicative subgroup of the finite fields. Ultimately, we show that all presented permutation polynomials are QM-inequivalent to the known ones.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102678"},"PeriodicalIF":1.2,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290830","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":"Several classes of linear codes with few weights derived from Weil sums","authors":"Zhao Hu ,&nbsp;Mingxiu Qiu ,&nbsp;Nian Li ,&nbsp;Xiaohu Tang ,&nbsp;Liwei Wu","doi":"10.1016/j.ffa.2025.102679","DOIUrl":"10.1016/j.ffa.2025.102679","url":null,"abstract":"<div><div>Linear codes with few weights have applications in secret sharing, authentication codes, association schemes and strongly regular graphs. In this paper, several classes of <em>t</em>-weight linear codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are presented with the defining sets given by the intersection, difference and union of two certain sets, where <span><math><mi>t</mi><mo>=</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>6</mn></math></span> and <em>q</em> is an odd prime power. By using Weil sums and Gauss sums, the parameters and weight distributions of these codes are determined completely. Moreover, three classes of optimal codes meeting the Griesmer bound are obtained, and computer experiments show that many (almost) optimal codes can be derived from our constructions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102679"},"PeriodicalIF":1.2,"publicationDate":"2025-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144290831","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}