Finite Fields and Their Applications最新文献

{"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}

{"title":"Transverse-freeness in finite geometries","authors":"Charlie Bruggemann ,&nbsp;Vera Choi ,&nbsp;Brian Freidin ,&nbsp;Jaedon Whyte","doi":"10.1016/j.ffa.2025.102663","DOIUrl":"10.1016/j.ffa.2025.102663","url":null,"abstract":"<div><div>We study the interplay between combinatorial and algebraic geometry via projective curves and hypersurfaces defined over a finite field that are tangent to every member of a class of low-degree varieties. Extending the 2-dimensional work of Asgarli, we first explore the lowest degrees attainable by smooth hypersurfaces in <em>n</em>-dimensional projective space that are tangent to every <em>k</em>-dimensional subspace, for some values of <em>n</em> and <em>k</em>. We then study projective surfaces that serve as models of finite inversive and hyperbolic planes, finite analogs of spherical and hyperbolic geometries. In these surfaces, we prove existence results for low-degree curves tangent to each of the lowest degree curves defined over the base field.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102663"},"PeriodicalIF":1.2,"publicationDate":"2025-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144203078","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":"Partial difference sets from unions of cyclotomic classes","authors":"Ka Hin Leung ,&nbsp;Koji Momihara ,&nbsp;Qing Xiang","doi":"10.1016/j.ffa.2025.102661","DOIUrl":"10.1016/j.ffa.2025.102661","url":null,"abstract":"<div><div>In their study of two-weight irreducible cyclic codes, Schmidt and White (2002) obtained a necessary and sufficient condition on <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> under which the multiplicative subgroup of index <em>N</em> of the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> forms a regular partial difference set (PDS) in the additive group of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. They also found 11 sporadic examples by a computer search aside from two known infinite families of PDS. In this paper, we study the problem of determining for which <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> a union of multiple cosets of the multiplicative subgroup of index <em>N</em> of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> forms a regular PDS in the additive group of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Building on the work of Schmidt and White, we find a necessary and sufficient numerical condition on the parameters <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>N</mi><mo>)</mo></math></span> for unions of multiple cyclotomic classes to form regular PDS in <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>,</mo><mo>+</mo><mo>)</mo></math></span>. We then apply the theorem to the situation where unions of a small number of classes are selected in a structured manner. We obtain a new infinite family of regular PDS not belonging to previously known families, and two sporadic examples of regular PDS (one of which is new) with the help of a computer research. We further propose a conjecture analogous to the Schmidt-White conjecture proposed in their 2002 paper.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102661"},"PeriodicalIF":1.2,"publicationDate":"2025-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144190116","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":"Two families of self-orthogonal codes with applications in LCD codes and optimally extendable codes","authors":"Dengcheng Xie,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102659","DOIUrl":"10.1016/j.ffa.2025.102659","url":null,"abstract":"<div><div>Self-orthogonal codes have interesting applications in quantum codes, linear complementary dual (LCD) codes and lattices. LCD codes and (almost) optimally extendable codes are useful to safeguard against Side-Channel Attacks (SCAs) and Fault Injection Attacks (FIAs). In this paper, we first give a lower bound of dual distances for augmented codes via the defining-set construction. Then we construct two families of <em>q</em>-ary self-orthogonal codes with determined weight distributions via defining-set construction and propose the parameters of their duals. Besides, several families of AMDS codes are obtained as byproducts, which are both length-optimal and dimension-optimal with respect to the Sphere-packing bound. As applications, these self-orthogonal codes are used to construct LCD codes and proved to be optimally extendable. As a consequence, our constructions produce some optimal codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102659"},"PeriodicalIF":1.2,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167720","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":"Almost Steiner systems in finite classical polar spaces","authors":"Yunxian Wu ,&nbsp;Tao Feng ,&nbsp;Lei Xu ,&nbsp;Menglong Zhang","doi":"10.1016/j.ffa.2025.102662","DOIUrl":"10.1016/j.ffa.2025.102662","url":null,"abstract":"<div><div>Let <span><math><mi>Q</mi></math></span> be a finite classical polar space with rank <em>n</em> and <span><math><mi>P</mi></math></span> be a collection of <em>k</em>-dimensional subspaces in <span><math><mi>Q</mi></math></span> called blocks. Let Λ be a set of nonnegative integers. A pair <span><math><mo>(</mo><mi>Q</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> is called a <em>t</em>-<span><math><msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mi>Λ</mi><mo>)</mo></mrow><mrow><mi>Q</mi></mrow></msub></math></span> design if every <em>t</em>-dimensional subspace in <span><math><mi>Q</mi></math></span> is contained in exactly <span><math><mi>λ</mi><mo>∈</mo><mi>Λ</mi></math></span> blocks of <span><math><mi>P</mi></math></span>. A <em>t</em>-<span><math><msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>Q</mi></mrow></msub></math></span> design is called a <em>Q</em>-Steiner system, which has <span><math><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>/</mo><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> blocks. Despite knowing very little about the existence of <em>Q</em>-Steiner systems, we demonstrate that given any positive integers <em>k</em> and <em>t</em> satisfying <span><math><mi>k</mi><mo>&gt;</mo><mi>t</mi></math></span>, for any finite polar space <span><math><mi>Q</mi></math></span> with a sufficiently large rank <em>n</em>, there exists a <em>t</em>-<span><math><msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>,</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo><mo>)</mo></mrow><mrow><mi>Q</mi></mrow></msub></math></span> design with <span><math><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>Q</mi></mrow></msub><mo>/</mo><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>k</mi></mtd></mtr><mtr><mtd><mi>t</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> blocks.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102662"},"PeriodicalIF":1.2,"publicationDate":"2025-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167721","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 from vectorial dual-bent functions","authors":"Zhicheng Wang ,&nbsp;Qiang Wang ,&nbsp;Shudi Yang","doi":"10.1016/j.ffa.2025.102660","DOIUrl":"10.1016/j.ffa.2025.102660","url":null,"abstract":"<div><div>Linear codes with few weights have wide applications in secret sharing, authentication codes, strongly regular graphs and association schemes. In this paper, we present linear codes from vectorial dual-bent functions and permutation polynomials, such that their parameters and weight distributions can be explicitly determined. In particular, some of them are three-weight optimal or almost optimal codes. As applications, we extend these codes to construct self-orthogonal codes and show the existence of asymmetric quantum codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102660"},"PeriodicalIF":1.2,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144154384","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":"Shortest-path and antichain metrics","authors":"Mahir Bilen Can,&nbsp;Dillon Montero","doi":"10.1016/j.ffa.2025.102658","DOIUrl":"10.1016/j.ffa.2025.102658","url":null,"abstract":"<div><div>In this paper, we introduce two new metrics for error-correcting codes that extend the classical Hamming metric. The first, called the shortest-path metric, coincides with the Niederreiter-Rosenbloom-Tsfasman (NRT) metric when the underlying poset is a disjoint union of equal-length chains. The second, called the antichain metric, is shown to align with the <em>b</em>-symbol Hamming weight under the same poset structure. We explore analogs of maximum distance separable (MDS) codes and perfect codes for both metrics and determine the corresponding weight enumerator polynomials. Additionally, we establish criteria for when the antichain metric yields one-weight perfect codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"108 ","pages":"Article 102658"},"PeriodicalIF":1.2,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134401","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}