Pub Date : 2025-01-03DOI: 10.1016/j.ffa.2024.102567
Tongliang Zhang , Lijing Zheng
Let . In this paper, we propose several classes of permutation pentanomials of the form over from some certain linearized polynomial by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials over for n satisfying are also constructed based on some bijections over the unit circle of with order .
{"title":"More classes of permutation pentanomials over finite fields with even characteristic","authors":"Tongliang Zhang , Lijing Zheng","doi":"10.1016/j.ffa.2024.102567","DOIUrl":"10.1016/j.ffa.2024.102567","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper, we propose several classes of permutation pentanomials of the form <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><mn>2</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>4</mn><mo>)</mo></math></span> from some certain linearized polynomial <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by using multivariate method and some techniques to determine the solutions of some equations. Furthermore, two classes of permutation pentanomials 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> for <em>n</em> satisfying <span><math><mn>3</mn><mo>|</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> are also constructed based on some bijections over the unit circle <span><math><msub><mrow><mi>U</mi></mrow><mrow><mi>τ</mi></mrow></msub></math></span> of <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 order <span><math><mi>τ</mi><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>q</mi><mo>+</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102567"},"PeriodicalIF":1.2,"publicationDate":"2025-01-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139958","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}
Pub Date : 2024-12-11DOI: 10.1016/j.ffa.2024.102547
Daniel Smith-Tone , Cristina Tone
Cheng et al. (2014) [6] introduced a substantial improvement to the Miura-Hashimoto-Takagi algorithm for solving sufficiently underdetermined systems of multivariate polynomial equations. This improvement claimed to make the algorithm polynomial time for instances satisfying , where m is the number of equations and n is the number of variables. While experimentally, the algorithm seems to work, we have uncovered a subtle error in the proof of time complexity for the algorithm. Due to the fact that there have been multiple proposals for algorithms based on this and related algorithms, as well as the recent submission to NIST's call for additional post-quantum digital signatures of a more modern “provably secure” version of the famous UOV digital signature algorithm based on the foundational structure of this algorithm, our observation may highlight a concerning theoretical deficiency in this area of research.
In this work, we provide a tight justification for the polynomial time complexity of the algorithm (with a very minor tweak), thereby justifying the complexity of enhancements based upon it as well. At the heart of this justification is a precise calculation of the probability of recovering a maximal depth path in polynomially many steps within a possibly exponentially large search tree. While this algorithmic problem is generic, we find that the parameters relevant for the application to the above algorithm are extremal and poorly studied. Thus, our analysis serves to clarify the boundary behavior of such search algorithms with respect to complexity classes.
{"title":"A correct justification for the CHMT algorithm for solving underdetermined multivariate systems","authors":"Daniel Smith-Tone , Cristina Tone","doi":"10.1016/j.ffa.2024.102547","DOIUrl":"10.1016/j.ffa.2024.102547","url":null,"abstract":"<div><div>Cheng et al. (2014) <span><span>[6]</span></span> introduced a substantial improvement to the Miura-Hashimoto-Takagi algorithm for solving sufficiently underdetermined systems of multivariate polynomial equations. This improvement claimed to make the algorithm polynomial time for instances satisfying <span><math><mi>n</mi><mo>≥</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable><mo>)</mo></mrow></math></span>, where <em>m</em> is the number of equations and <em>n</em> is the number of variables. While experimentally, the algorithm seems to work, we have uncovered a subtle error in the proof of time complexity for the algorithm. Due to the fact that there have been multiple proposals for algorithms based on this and related algorithms, as well as the recent submission to NIST's call for additional post-quantum digital signatures of a more modern “provably secure” version of the famous UOV digital signature algorithm based on the foundational structure of this algorithm, our observation may highlight a concerning theoretical deficiency in this area of research.</div><div>In this work, we provide a tight justification for the polynomial time complexity of the algorithm (with a very minor tweak), thereby justifying the complexity of enhancements based upon it as well. At the heart of this justification is a precise calculation of the probability of recovering a maximal depth path in polynomially many steps within a possibly exponentially large search tree. While this algorithmic problem is generic, we find that the parameters relevant for the application to the above algorithm are extremal and poorly studied. Thus, our analysis serves to clarify the boundary behavior of such search algorithms with respect to complexity classes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102547"},"PeriodicalIF":1.2,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139956","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}
Error correcting codes have recently gained more attention due to their applications in quantum resistant cryptography. Their suitability depends on their indistinguishability from random codes. In that sense, the study of the square code of a particular code provides a tool for distinguishing random codes from not random ones.
With this motivation, the square codes of some semisimple bilateral group codes, as abelian and dihedral ones, are studied in this paper. For this purpose, bilateral group codes are described as evaluation codes by means of the absolutely irreducible characters of the group. Finally, some results on self-duality and self-orthogonality are recovered under this alternative point of view.
{"title":"On the square code of group codes","authors":"Alejandro Piñera Nicolás , Ignacio Fernández Rúa , Adriana Suárez Corona","doi":"10.1016/j.ffa.2024.102548","DOIUrl":"10.1016/j.ffa.2024.102548","url":null,"abstract":"<div><div>Error correcting codes have recently gained more attention due to their applications in quantum resistant cryptography. Their suitability depends on their indistinguishability from random codes. In that sense, the study of the square code of a particular code provides a tool for distinguishing random codes from not random ones.</div><div>With this motivation, the square codes of some semisimple bilateral group codes, as abelian and dihedral ones, are studied in this paper. For this purpose, bilateral group codes are described as evaluation codes by means of the absolutely irreducible characters of the group. Finally, some results on self-duality and self-orthogonality are recovered under this alternative point of view.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"103 ","pages":"Article 102548"},"PeriodicalIF":1.2,"publicationDate":"2024-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143139957","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}
Pub Date : 2024-11-27DOI: 10.1016/j.ffa.2024.102546
Alp Bassa , Ricardo Menares
We give iterative constructions for irreducible polynomials over of degree for all , starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions . The R-transform introduced by Cohen is recovered as a particular case corresponding to , hence we obtain a generalization of Cohen's R-transform () to arbitrary degrees . Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of we recover and generalize a recursive construction of Panario, Reis and Wang.
我们从 n 度的不可还原多项式出发，给出了所有 r≥0 的 n⋅tr Fq 上不可还原多项式的迭代构造。科恩引入的 R 变换作为与 x2 相对应的特殊情况被复原，因此我们得到了科恩 R 变换 (t=2) 对任意度 t≥2 的推广。从与适当的合一根相乘的不变性推导出了自还原性和根在某些自动形态下的不变性等重要性质。扩展到 Fq 的二次扩展，我们恢复并推广了帕纳里奥、雷斯和王的递归构造。
{"title":"The R-transform as power map and its generalizations to higher degree","authors":"Alp Bassa , Ricardo Menares","doi":"10.1016/j.ffa.2024.102546","DOIUrl":"10.1016/j.ffa.2024.102546","url":null,"abstract":"<div><div>We give iterative constructions for irreducible polynomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> of degree <span><math><mi>n</mi><mo>⋅</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> for all <span><math><mi>r</mi><mo>≥</mo><mn>0</mn></math></span>, starting from irreducible polynomials of degree <em>n</em>. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup></math></span>. The <em>R</em>-transform introduced by Cohen is recovered as a particular case corresponding to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, hence we obtain a generalization of Cohen's <em>R</em>-transform (<span><math><mi>t</mi><mo>=</mo><mn>2</mn></math></span>) to arbitrary degrees <span><math><mi>t</mi><mo>≥</mo><mn>2</mn></math></span>. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> we recover and generalize a recursive construction of Panario, Reis and Wang.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102546"},"PeriodicalIF":1.2,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722321","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}
Pub Date : 2024-11-26DOI: 10.1016/j.ffa.2024.102544
Nour Alnajjarine , Michel Lavrauw
This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in , for q odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in , both for q odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in . The main contribution is the determination of the distribution of the different types of hyperplanes incident with the K-orbit representatives of points and lines of , where , is the subgroup of stabilizing the Veronese surface.
本文是对 L. E. Dickson 于 1908 年通过研究 PG(2,q) 中圆锥的铅笔的投影等价类（q 为奇数）而提出的有限域上圆锥的线性系统分类这一长期未决问题的一个解决方案的贡献。本文为 PG(2,q) 中圆锥的网状和方形的投影等价类确定了一组完整的不变式，无论是 q 为奇数还是偶数。我们的方法主要是几何方法，涉及对 PG(5,q) 中维罗尼斯曲面的几何和组合性质的全面研究。我们的主要贡献是确定了与 PG(5,q) 的 K 轨道代表点和线相关的不同类型超平面的分布，其中 K≅PGL(3,q) 是 PGL(6,q) 的子群，稳定了 Veronese 曲面。
{"title":"Webs and squabs of conics over finite fields","authors":"Nour Alnajjarine , Michel Lavrauw","doi":"10.1016/j.ffa.2024.102544","DOIUrl":"10.1016/j.ffa.2024.102544","url":null,"abstract":"<div><div>This paper is a contribution towards a solution for the longstanding open problem of classifying linear systems of conics over finite fields initiated by L. E. Dickson in 1908, through his study of the projective equivalence classes of pencils of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, for <em>q</em> odd. In this paper a set of complete invariants is determined for the projective equivalence classes of webs and of squabs of conics in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, both for <em>q</em> odd and even. Our approach is mainly geometric, and involves a comprehensive study of the geometric and combinatorial properties of the Veronese surface in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. The main contribution is the determination of the distribution of the different types of hyperplanes incident with the <em>K</em>-orbit representatives of points and lines of <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>5</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, where <span><math><mi>K</mi><mo>≅</mo><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>, is the subgroup of <span><math><mrow><mi>PGL</mi></mrow><mo>(</mo><mn>6</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> stabilizing the Veronese surface.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102544"},"PeriodicalIF":1.2,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722322","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}
Pub Date : 2024-11-21DOI: 10.1016/j.ffa.2024.102545
Jin Li , Huilian Zhu , Shixin Zhu
In this paper, we study some q-ary λ-constacyclic BCH codes of length with some large designed distances for and respectively, where q is an odd prime power and . The dimensions and the lower bounds on the minimum distances of these codes are given by using recurrence relations and the introduced definitions of sequences. The code examples presented in this paper indicate that these codes have good parameters in general.
{"title":"Some q-ary constacyclic BCH codes with length qm+12","authors":"Jin Li , Huilian Zhu , Shixin Zhu","doi":"10.1016/j.ffa.2024.102545","DOIUrl":"10.1016/j.ffa.2024.102545","url":null,"abstract":"<div><div>In this paper, we study some <em>q</em>-ary <em>λ</em>-constacyclic BCH codes of length <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> with some large designed distances for <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>=</mo><mn>2</mn></math></span> and <span><math><mn>2</mn><mo>+</mo><mo>⌈</mo><mfrac><mrow><mn>4</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>3</mn></mrow></mfrac><mo>⌉</mo><mo>≤</mo><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>≤</mo><mi>q</mi><mo>−</mo><mn>1</mn></math></span> respectively, where <em>q</em> is an odd prime power and <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><mi>λ</mi><mo>)</mo><mo>|</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>. The dimensions and the lower bounds on the minimum distances of these codes are given by using recurrence relations and the introduced definitions of sequences. The code examples presented in this paper indicate that these codes have good parameters in general.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"102 ","pages":"Article 102545"},"PeriodicalIF":1.2,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142707366","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}
Pub Date : 2024-11-14DOI: 10.1016/j.ffa.2024.102542
Qi Zhang , Weiqiong Wang , Shuyu Luo , Yue Li
For different odd primes k, l, p, and positive integers s, m, n, the polynomial in is explicitly factorized, where p is the characteristic of , . All repeated-root constacyclic codes and their dual codes of length over are characterized. In addition, the characterization and enumeration of all linear complementary dual (LCD) cyclic and negacyclic codes of length over are obtained.
{"title":"Repeated-root constacyclic codes of length kslmpn over finite fields","authors":"Qi Zhang , Weiqiong Wang , Shuyu Luo , Yue Li","doi":"10.1016/j.ffa.2024.102542","DOIUrl":"10.1016/j.ffa.2024.102542","url":null,"abstract":"<div><div>For different odd primes <em>k</em>, <em>l</em>, <em>p</em>, and positive integers <em>s</em>, <em>m</em>, <em>n</em>, the polynomial <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msup><mo>−</mo><mi>λ</mi></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></math></span> is explicitly factorized, where <em>p</em> is the characteristic of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, <span><math><mi>λ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. All repeated-root constacyclic codes and their dual codes of length <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are characterized. In addition, the characterization and enumeration of all linear complementary dual (LCD) cyclic and negacyclic codes of length <span><math><msup><mrow><mi>k</mi></mrow><mrow><mi>s</mi></mrow></msup><msup><mrow><mi>l</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> are obtained.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102542"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660203","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}
Pub Date : 2024-11-14DOI: 10.1016/j.ffa.2024.102543
Nikolai S. Nadirashvili , Michael A. Tsfasman
We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman–Vlăduţ theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P. Serre, using results of R.M. Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for example, the Lebesgue measure on a segment). Here we settle Serre's problem proving that Serre conditions are sufficient, and thus describe completely the set of measures corresponding to abelian varieties.
{"title":"Complete description of measures corresponding to Abelian varieties over finite fields","authors":"Nikolai S. Nadirashvili , Michael A. Tsfasman","doi":"10.1016/j.ffa.2024.102543","DOIUrl":"10.1016/j.ffa.2024.102543","url":null,"abstract":"<div><div>We study probability measures corresponding to families of abelian varieties over a finite field. These measures play an important role in the Tsfasman–Vlăduţ theory of asymptotic zeta-functions defining completely the limit zeta-function of the family. J.-P. Serre, using results of R.M. Robinson on conjugate algebraic integers, described the possible set of measures than can correspond to families of abelian varieties over a finite field. The problem whether all such measures actually occur was left open. Moreover, Serre supposed that not all such measures correspond to abelian varieties (for example, the Lebesgue measure on a segment). Here we settle Serre's problem proving that Serre conditions are sufficient, and thus describe completely the set of measures corresponding to abelian varieties.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102543"},"PeriodicalIF":1.2,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660194","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}
Pub Date : 2024-11-07DOI: 10.1016/j.ffa.2024.102541
Yun Fan, Yue Leng
In this paper, we investigate the existence and asymptotic properties of self-dual 2-quasi negacyclic codes of length 2n over a finite field of cardinality q. When n is odd, we show that the q-ary self-dual 2-quasi negacyclic codes exist if and only if . When n is even, we prove that the q-ary self-dual 2-quasi negacyclic codes always exist. By using the technique introduced in this paper, we prove that q-ary self-dual 2-quasi negacyclic codes are asymptotically good.
当 n 为奇数时，我们证明了当且仅当 q≢-1(mod4)时 q-ary 自双 2-quasi 负环码存在。当 n 为偶数时，我们证明 q-ary 自双 2-quasi 负环码总是存在的。利用本文介绍的技术，我们证明了 qary 自双 2-quasi 负环码是渐近良好的。
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Pub Date : 2024-11-07DOI: 10.1016/j.ffa.2024.102540
Nika Salia , Dávid Tóth
This paper demonstrates an analog of the Erdős–Ko–Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins.
A k-uniform family of subsets of a set of size n is ℓ-intersecting if any two subsets in the family intersect in at least ℓ elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erdős–Ko–Rado theorem, which establishes a sharp upper bound on the size of these families. Here, we extend the Erdős–Ko–Rado theorem to polynomial rings over finite fields.
Specifically, we determine the largest possible size of a family of monic polynomials, each of degree n, over a finite field , where every pair of polynomials in the family shares a common factor of degree at least ℓ. We prove that the upper bound for this size is and characterize all extremal families that achieve this maximum size.
Extending to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least ℓ, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most n, we affirm that only trivial families achieve the corresponding upper bound.
本文证明了厄尔多-柯-拉多定理在有限域上多项式环中的类比，肯定地回答了汤普金斯（C. Tompkins）的猜想。如果一个大小为 n 的集合的 k 个均匀子集族中的任意两个子集至少有 ℓ 个元素相交，那么这个子集族就是 ℓ 相交族。对这种相交族的研究是极值集合理论的核心课题，其根源可追溯到 1961 年的开创性厄多-柯-拉多定理，该定理为这些族的大小确定了一个尖锐的上限。在这里，我们将厄尔多斯-柯-拉多定理扩展到有限域上的多项式环。具体地说，我们确定了有限域 Fq 上每个度数为 n 的单项式族的最大可能规模，其中族中的每对多项式都有一个度数至少为 ℓ 的公因子。我们证明了这一大小的上界是 qn-ℓ，并描述了达到这一最大大小的所有极值族的特征。扩展到三重相交族，其中每个三重多项式共享至少 ℓ 阶的公因子，我们证明只有三重族达到了相应的上界。此外，通过将条件放宽到包括阶数至多为 n 的多项式，我们肯定只有三交系达到了相应的上限。
{"title":"Intersecting families of polynomials over finite fields","authors":"Nika Salia , Dávid Tóth","doi":"10.1016/j.ffa.2024.102540","DOIUrl":"10.1016/j.ffa.2024.102540","url":null,"abstract":"<div><div>This paper demonstrates an analog of the Erdős–Ko–Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins.</div><div>A <em>k</em>-uniform family of subsets of a set of size <em>n</em> is <em>ℓ</em>-intersecting if any two subsets in the family intersect in at least <em>ℓ</em> elements. The study of such intersecting families is a core subject of extremal set theory, tracing its roots to the seminal 1961 Erdős–Ko–Rado theorem, which establishes a sharp upper bound on the size of these families. Here, we extend the Erdős–Ko–Rado theorem to polynomial rings over finite fields.</div><div>Specifically, we determine the largest possible size of a family of monic polynomials, each of degree <em>n</em>, over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where every pair of polynomials in the family shares a common factor of degree at least <em>ℓ</em>. We prove that the upper bound for this size is <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>ℓ</mi></mrow></msup></math></span> and characterize all extremal families that achieve this maximum size.</div><div>Extending to triple-intersecting families, where every triplet of polynomials shares a common factor of degree at least <em>ℓ</em>, we prove that only trivial families achieve the corresponding upper bound. Moreover, by relaxing the conditions to include polynomials of degree at most <em>n</em>, we affirm that only trivial families achieve the corresponding upper bound.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"101 ","pages":"Article 102540"},"PeriodicalIF":1.2,"publicationDate":"2024-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142660192","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}
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