Finite Fields and Their Applications最新文献

{"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}

{"title":"Multispreads","authors":"Denis S. Krotov,&nbsp;Ivan Yu. Mogilnykh","doi":"10.1016/j.ffa.2025.102675","DOIUrl":"10.1016/j.ffa.2025.102675","url":null,"abstract":"<div><div>Additive one-weight codes over a finite field of non-prime order are equivalent to special subspace coverings of the points of a projective space, which we call multispreads. The current paper is devoted to the characterization of the parameters of multispreads, which is equivalent to the characterization of the parameters of additive one-weight codes and, via duality, of additive completely regular codes of covering radius 1 (intriguing sets). We characterize these parameters for the case of the prime-square order of the field and make a partial characterization for the prime-cube case and the case of the fourth degree of a prime, including a complete characterization for orders 8, 27, and 16.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102675"},"PeriodicalIF":1.2,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272582","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":"Upper bounds for L-functions in positive characteristic","authors":"Si-Han Liu ,&nbsp;Jia-Yan Yao","doi":"10.1016/j.ffa.2025.102677","DOIUrl":"10.1016/j.ffa.2025.102677","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field with <em>q</em> elements, and <em>K</em> an algebraic function field over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> as its field of constants. In this work we shall establish the upper bounds for the zeta function <span><math><msub><mrow><mi>ζ</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>)</mo></math></span> and the <em>L</em>-function <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>K</mi></mrow></msub><mo>(</mo><mi>s</mi><mo>,</mo><mi>χ</mi><mo>)</mo></math></span> with <span><math><mn>0</mn><mo>&lt;</mo><mrow><mi>Re</mi></mrow><mo>(</mo><mi>s</mi><mo>)</mo><mo>&lt;</mo><mn>1</mn></math></span>, where <em>χ</em> is a Hecke character over <em>K</em>. In particular, for <span><math><mrow><mi>Re</mi></mrow><mo>(</mo><mi>s</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>/</mo><mn>2</mn></math></span>, we obtain an upper bound of the Lindelöf's type.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102677"},"PeriodicalIF":1.2,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144272583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The compositional inverses of permutation polynomials of the form ∑i=1kbi(xpm+x+δ)si−x over Fp2m","authors":"Danyao Wu ,&nbsp;Pingzhi Yuan ,&nbsp;Huanhuan Guan ,&nbsp;Juan Li","doi":"10.1016/j.ffa.2025.102681","DOIUrl":"10.1016/j.ffa.2025.102681","url":null,"abstract":"<div><div>In this paper, we present the compositional inverses of several classes permutation polynomials of the form <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>δ</mi><mo>)</mo></mrow><mrow><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msup><mo>−</mo><mi>x</mi></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>, where for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span>, <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>m</mi></math></span> are positive integers, <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>δ</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>m</mi></mrow></msup></mrow></msub></math></span>, and <em>p</em> is prime.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102681"},"PeriodicalIF":1.2,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144261668","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":"Dynamical irreducibility of certain families of polynomials over finite fields","authors":"Tori Day ,&nbsp;Rebecca DeLand ,&nbsp;Jamie Juul ,&nbsp;Cigole Thomas ,&nbsp;Bianca Thompson ,&nbsp;Bella Tobin","doi":"10.1016/j.ffa.2025.102666","DOIUrl":"10.1016/j.ffa.2025.102666","url":null,"abstract":"<div><div>We determine necessary and sufficient conditions for unicritical polynomials conjugate to <span><math><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>+</mo><mi>c</mi></math></span> to be dynamically irreducible over finite fields. This result extends the results of Boston-Jones and Hamblen-Jones-Madhu regarding the dynamical irreducibility of particular families of unicritical polynomials. We also investigate dynamical irreducibility conditions for cubic and shifted linearized polynomials.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102666"},"PeriodicalIF":1.2,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144261667","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 up to Hartmann–Tzeng and Roos bounds for rank codes","authors":"José Manuel Muñoz","doi":"10.1016/j.ffa.2025.102676","DOIUrl":"10.1016/j.ffa.2025.102676","url":null,"abstract":"<div><div>A class of linear block codes which simultaneously generalizes Gabidulin codes and a class of skew cyclic codes is defined. For these codes, both a Hartmann–Tzeng-like bound and a Roos-like bound, with respect to their rank distance, are described, and corresponding nearest-neighbor decoding algorithms are presented. Additional necessary conditions so that decoding can be done up to the described bounds are studied. Subfield subcodes and interleaved codes from the considered class of codes are also described, since they allow an unbounded length for the codes, providing a decoding algorithm for them; additionally, both approaches are shown to yield equivalent codes with respect to the rank metric.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102676"},"PeriodicalIF":1.2,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144253583","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":"Construction of Galois self-orthogonal MDS codes with larger dimensions","authors":"Ruhao Wan,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102665","DOIUrl":"10.1016/j.ffa.2025.102665","url":null,"abstract":"&lt;div&gt;&lt;div&gt;Let &lt;span&gt;&lt;math&gt;&lt;mi&gt;q&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;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; be a prime power, &lt;em&gt;e&lt;/em&gt; be an integer with &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;gcd&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;. In this paper, for a vector &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;msup&gt;&lt;mrow&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;mi&gt;q&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;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; and a &lt;em&gt;q&lt;/em&gt;-ary linear code &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, we give some necessary and sufficient conditions for the equivalent code &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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 &lt;span&gt;&lt;math&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; and the extended code of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;Φ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&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; to be &lt;em&gt;e&lt;/em&gt;-Galois self-orthogonal. We then directly obtain some necessary and sufficient conditions for (extended) generalized Reed-Solomon (GRS and EGRS) codes to be &lt;em&gt;e&lt;/em&gt;-Galois self-orthogonal. From this we show that if &lt;span&gt;&lt;math&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mi&gt;min&lt;/mi&gt;&lt;mo&gt;⁡&lt;/mo&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mi&gt;max&lt;/mi&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;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⌈&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌉&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;max&lt;/mi&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;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;⌈&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;n&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;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&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;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;⌉&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, there is no &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;[&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;]&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;q&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; &lt;em&gt;e&lt;/em&gt;-Galois self-orthogonal (extended) GRS code. Furthermore, for all possible &lt;em&gt;e&lt;/em&gt; satisfying &lt;span&gt;&lt;math&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;≤&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;, we classify them into three cases (1) &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; odd and &lt;em&gt;p&lt;/em&gt; even; (2) &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; odd and &lt;em&gt;p&lt;/em&gt; odd; (3) &lt;span&gt;&lt;math&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt; even, and construct several new classes of &lt;em&gt;e&lt;/em&gt;-Galois self-orthogonal maximum distance separable (MDS) codes. It is worth noting that our &lt;em&gt;e&lt;/em&gt;-Galois self-orthogonal MDS","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102665"},"PeriodicalIF":1.2,"publicationDate":"2025-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144222654","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}