Pub Date : 2026-03-01Epub Date: 2025-11-25DOI: 10.1016/j.ffa.2025.102761
Li Zhu , Jinle Liu , Hongfeng Wu
Cyclotomic coset is a classical notion in the theory of finite fields which has wide applications in various computation problems. Let q be a prime power, and n be a positive integer coprime to q. In this paper we determine explicitly the representatives and the sizes of all q-cyclotomic cosets modulo n in the general settings. We introduce the definition of 2-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the 2-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of q-cyclotomic cosets modulo n, we improve the formulas for the factorizations of and of over given in [4]. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated.
{"title":"Explicit representatives and sizes of cyclotomic cosets and their application to cyclic codes over finite fields","authors":"Li Zhu , Jinle Liu , Hongfeng Wu","doi":"10.1016/j.ffa.2025.102761","DOIUrl":"10.1016/j.ffa.2025.102761","url":null,"abstract":"<div><div>Cyclotomic coset is a classical notion in the theory of finite fields which has wide applications in various computation problems. Let <em>q</em> be a prime power, and <em>n</em> be a positive integer coprime to <em>q</em>. In this paper we determine explicitly the representatives and the sizes of all <em>q</em>-cyclotomic cosets modulo <em>n</em> in the general settings. We introduce the definition of 2-adic cyclotomic system, which is a profinite space consists of certain compatible sequences of cyclotomic cosets. A precise characterization of the structure of the 2-adic cyclotomic system is given, which reveals the general formula for representatives of cyclotomic cosets. With the representatives and the sizes of <em>q</em>-cyclotomic cosets modulo <em>n</em>, we improve the formulas for the factorizations of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> and of <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> given in <span><span>[4]</span></span>. As a consequence, we classify the cyclic codes over finite fields via giving their generator polynomials. Moreover, the self-dual cyclic codes are determined and enumerated.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102761"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624202","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 : 2026-03-01Epub Date: 2025-12-16DOI: 10.1016/j.ffa.2025.102780
H. Navarro, Luz A. Pérez
In this paper, we present two methods for constructing curves of Kummer type with many rational points over finite fields. The first method is based on binomials, while the second employs reciprocal polynomials. The latter is an extension of the method introduced by Gupta et al. (2023) [19] over quadratic finite fields, to non-prime finite fields. As a result, we found 63 new records and 37 new entries for the online table of curves with many points found at manYPoints.
{"title":"New curves of Kummer type with many rational points over finite fields","authors":"H. Navarro, Luz A. Pérez","doi":"10.1016/j.ffa.2025.102780","DOIUrl":"10.1016/j.ffa.2025.102780","url":null,"abstract":"<div><div>In this paper, we present two methods for constructing curves of Kummer type with many rational points over finite fields. The first method is based on binomials, while the second employs reciprocal polynomials. The latter is an extension of the method introduced by Gupta et al. (2023) <span><span>[19]</span></span> over quadratic finite fields, to non-prime finite fields. As a result, we found 63 new records and 37 new entries for the online table of curves with many points found at <span><span>manYPoints</span><svg><path></path></svg></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102780"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789554","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 : 2026-03-01Epub Date: 2025-12-15DOI: 10.1016/j.ffa.2025.102784
Mariusz Kwiatkowski, Mark Pankov, Adam Tyc
Consider the point-line geometry whose maximal singular subspaces correspond to q-ary simplex codes of dimension k. Maximal cliques in the collinearity graph of this geometry contain no more than elements and maximal singular subspaces of are n-cliques of this graph. If , then and there is a one-to-one correspondence between -cliques of the collinearity graph and symmetric -designs. For the case when we construct a class of n-cliques distinct from maximal singular subspaces. In the case when , some of these cliques are normal rational curves.
{"title":"One class of maximal cliques in the collinearity graphs of geometries related to simplex codes","authors":"Mariusz Kwiatkowski, Mark Pankov, Adam Tyc","doi":"10.1016/j.ffa.2025.102784","DOIUrl":"10.1016/j.ffa.2025.102784","url":null,"abstract":"<div><div>Consider the point-line geometry <span><math><mi>S</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> whose maximal singular subspaces correspond to <em>q</em>-ary simplex codes of dimension <em>k</em>. Maximal cliques in the collinearity graph of this geometry contain no more than <span><math><mi>n</mi><mo>=</mo><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> elements and maximal singular subspaces of <span><math><mi>S</mi><mo>(</mo><mi>k</mi><mo>,</mo><mi>q</mi><mo>)</mo></math></span> are <em>n</em>-cliques of this graph. If <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>, then <span><math><mi>n</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> and there is a one-to-one correspondence between <span><math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-cliques of the collinearity graph and symmetric <span><math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span>-designs. For the case when <span><math><mi>q</mi><mo>≥</mo><mn>5</mn></math></span> we construct a class of <em>n</em>-cliques distinct from maximal singular subspaces. In the case when <span><math><mi>k</mi><mo>=</mo><mn>2</mn></math></span>, some of these cliques are normal rational curves.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102784"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789556","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 : 2026-03-01Epub Date: 2025-12-11DOI: 10.1016/j.ffa.2025.102781
Tetsushi Ito , Ren Tatematsu , Takahiro Tsushima
The van der Geer–van der Vlugt curves are Artin–Schreier coverings of the affine line defined by linearized polynomials over finite fields. We provide several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse–Weil inequalities. As applications, we identify several maximal (or minimal) curves within this family. Our proofs are based on an explicit formula for the L-polynomials, recently obtained by Takeuchi and the third author.
van der Geer-van der Vlugt曲线是有限域上由线性化多项式定义的仿射线的Artin-Schreier覆盖。我们给出了它们最大或最小的几个标准，即达到Hasse-Weil不等式的上界或下界。作为应用，我们在这个族中确定了几个最大（或最小）曲线。我们的证明是基于最近由Takeuchi和第三作者获得的l多项式的显式公式。
{"title":"Criteria of maximality and minimality of van der Geer–van der Vlugt curves","authors":"Tetsushi Ito , Ren Tatematsu , Takahiro Tsushima","doi":"10.1016/j.ffa.2025.102781","DOIUrl":"10.1016/j.ffa.2025.102781","url":null,"abstract":"<div><div>The van der Geer–van der Vlugt curves are Artin–Schreier coverings of the affine line defined by linearized polynomials over finite fields. We provide several criteria for them to be maximal or minimal, i.e. attaining the upper or lower bound in the Hasse–Weil inequalities. As applications, we identify several maximal (or minimal) curves within this family. Our proofs are based on an explicit formula for the <em>L</em>-polynomials, recently obtained by Takeuchi and the third author.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102781"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145737313","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 : 2026-03-01Epub Date: 2025-11-21DOI: 10.1016/j.ffa.2025.102762
John Bamberg , Geertrui Van de Voorde
The Hermitian Veronesean in , given by , is a well-studied rational curve, and forms a special set of the Hermitian surface . In this paper, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.
{"title":"On the Hermitian Veronesean","authors":"John Bamberg , Geertrui Van de Voorde","doi":"10.1016/j.ffa.2025.102762","DOIUrl":"10.1016/j.ffa.2025.102762","url":null,"abstract":"<div><div>The Hermitian Veronesean in <span><math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>3</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, given by <span><math><mi>V</mi><mo>:</mo><mo>=</mo><mo>{</mo><mo>(</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>:</mo><mi>x</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>}</mo><mo>∪</mo><mo>{</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo><mo>}</mo></math></span>, is a well-studied rational curve, and forms a <em>special</em> set of the Hermitian surface <span><math><mi>H</mi><mo>(</mo><mn>3</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. In this paper, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102762"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580235","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 : 2026-03-01Epub Date: 2025-11-26DOI: 10.1016/j.ffa.2025.102763
Krishna Kaipa , Puspendu Pradhan
We consider the problem of classifying the lines of the projective 3-space over a finite field into orbits of the group of linear symmetries of the twisted cubic C. The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into -orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.
{"title":"On the PGL2(q)-orbits of lines of PG(3,q) and binary quartic forms in characteristic three","authors":"Krishna Kaipa , Puspendu Pradhan","doi":"10.1016/j.ffa.2025.102763","DOIUrl":"10.1016/j.ffa.2025.102763","url":null,"abstract":"<div><div>We consider the problem of classifying the lines of the projective 3-space <span><math><mi>P</mi><mi>G</mi><mo>(</mo><mn>3</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> into orbits of the group <span><math><mi>P</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> of linear symmetries of the twisted cubic <em>C</em>. The problem has been solved in literature in characteristic different from 3, and in this work, we solve the problem in characteristic 3. We reduce this problem to another problem, which is the classification of binary quartic forms into <span><math><mi>P</mi><mi>G</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-orbits. We first solve the latter problem and use to solve the former problem. We also obtain the point-line and the line-plane incidence structures of the point, line, and plane orbits.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102763"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145624274","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 : 2026-03-01Epub Date: 2026-01-02DOI: 10.1016/j.ffa.2025.102787
Wei Tang , Yue Zhou
Let denote the set of symmetric bilinear forms over an n-dimensional -vector space. A subset of is called a d-code if the rank of is larger than or equal to d for any distinct A and B in . If is further closed under matrix addition, then is sharply upper bounded by if is even and if is odd. Additive codes meeting these upper bounds are called maximum. There are very few known constructions of them. In this paper, we obtain a new family of maximum -linear -codes in for and 10 which are not equivalent to any known constructions. Furthermore, we completely determine the equivalence between distinct members in this new family.
{"title":"A new family of maximum linear symmetric rank-distance codes","authors":"Wei Tang , Yue Zhou","doi":"10.1016/j.ffa.2025.102787","DOIUrl":"10.1016/j.ffa.2025.102787","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> denote the set of symmetric bilinear forms over an <em>n</em>-dimensional <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-vector space. A subset <span><math><mi>C</mi></math></span> of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> is called a <em>d</em>-code if the rank of <span><math><mi>A</mi><mo>−</mo><mi>B</mi></math></span> is larger than or equal to <em>d</em> for any distinct <em>A</em> and <em>B</em> in <span><math><mi>C</mi></math></span>. If <span><math><mi>C</mi></math></span> is further closed under matrix addition, then <span><math><mo>|</mo><mi>C</mi><mo>|</mo></math></span> is sharply upper bounded by <span><math><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi><mo>(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></msup></math></span> if <span><math><mi>n</mi><mo>−</mo><mi>d</mi></math></span> is even and <span><math><msup><mrow><mi>q</mi></mrow><mrow><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>n</mi><mo>−</mo><mi>d</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow></msup></math></span> if <span><math><mi>n</mi><mo>−</mo><mi>d</mi></math></span> is odd. Additive codes meeting these upper bounds are called maximum. There are very few known constructions of them. In this paper, we obtain a new family of maximum <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>-linear <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>-codes in <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>=</mo><mn>6</mn><mo>,</mo><mn>8</mn></math></span> and 10 which are not equivalent to any known constructions. Furthermore, we completely determine the equivalence between distinct members in this new family.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102787"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883540","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 : 2026-03-01Epub Date: 2025-12-15DOI: 10.1016/j.ffa.2025.102778
Yajing Zhou , Rongquan Feng
Let be the ring of residue classes modulo n, and let be the group of its units. In 2017, Mollahajiaghaei presented a formula for the number of solutions of the congruence . This paper considers the addition of squares and cubes over . Specifically, when n is a prime number such that , we correct the formula given by Mollahajiaghaei.
{"title":"On the addition of squares and cubes of units modulo n","authors":"Yajing Zhou , Rongquan Feng","doi":"10.1016/j.ffa.2025.102778","DOIUrl":"10.1016/j.ffa.2025.102778","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the ring of residue classes modulo <em>n</em>, and let <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> be the group of its units. In 2017, Mollahajiaghaei presented a formula for the number of solutions <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo><mo>∈</mo><msup><mrow><mo>(</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup></math></span> of the congruence <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msubsup><mo>≡</mo><mi>c</mi><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>n</mi><mo>)</mo></math></span>. This paper considers the addition of squares and cubes over <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. Specifically, when <em>n</em> is a prime number such that <span><math><mi>n</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math></span>, we correct the formula given by Mollahajiaghaei.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102778"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145789555","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 : 2026-03-01Epub Date: 2025-11-20DOI: 10.1016/j.ffa.2025.102753
Guido Lido
We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially follow the heuristically quasi-polynomial algorithm presented by Barbulescu, Gaudry, Joux and Thomé. The main difference is to use a presentation of the finite field based on elliptic curves: the abundance of elliptic curves ensures the existence of such a presentation.
{"title":"A provably quasi-polynomial algorithm for the discrete logarithm problem in finite fields of small characteristic","authors":"Guido Lido","doi":"10.1016/j.ffa.2025.102753","DOIUrl":"10.1016/j.ffa.2025.102753","url":null,"abstract":"<div><div>We describe a provably quasi-polynomial algorithm to compute discrete logarithms in the multiplicative groups of finite fields of small characteristic, that is finite fields whose characteristic is logarithmic in the order. We partially follow the heuristically quasi-polynomial algorithm presented by Barbulescu, Gaudry, Joux and Thomé. The main difference is to use a presentation of the finite field based on elliptic curves: the abundance of elliptic curves ensures the existence of such a presentation.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102753"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145546710","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 : 2026-03-01Epub Date: 2025-12-30DOI: 10.1016/j.ffa.2025.102786
Xiaoran Wang, Junling Zhou
This paper concentrates on constructing infinite families of non-simple subspace 2-designs and 3-designs with block dimension 4. We investigate in detail the structure of the -incidence matrix between 2-subspaces and 4-subspaces of with . Employing the incidence matrix, we establish two recursive constructions for 2- designs, which are based on a 2- design and a 2- design, respectively. Several new infinite classes of simple q-analogs of group divisible designs (q-GDDs) with block dimension 4 are also produced. Making use of the recursive constructions and new q-GDDs, plenty of new infinite series of non-simple subspace 2-designs with block dimension 4 are constructed. We also study the -incidence matrix between 3-subspaces and 4-subspaces. From this, a recursive construction and a new infinite family of non-simple 3- designs are produced as well.
{"title":"Infinite families of non-simple subspace 2- and 3-designs with block dimension 4","authors":"Xiaoran Wang, Junling Zhou","doi":"10.1016/j.ffa.2025.102786","DOIUrl":"10.1016/j.ffa.2025.102786","url":null,"abstract":"<div><div>This paper concentrates on constructing infinite families of non-simple subspace 2-designs and 3-designs with block dimension 4. We investigate in detail the structure of the <span><math><mrow><mi>GL</mi></mrow><mo>(</mo><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></math></span>-incidence matrix between 2-subspaces and 4-subspaces of <span><math><mi>GF</mi><mspace></mspace><msup><mrow><mo>(</mo><mi>q</mi><mo>)</mo></mrow><mrow><mi>m</mi><mi>l</mi></mrow></msup></math></span> with <span><math><mi>m</mi><mo>,</mo><mi>l</mi><mo>≥</mo><mn>3</mn></math></span>. Employing the incidence matrix, we establish two recursive constructions for 2-<span><math><msub><mrow><mo>(</mo><mi>m</mi><mi>l</mi><mo>,</mo><mn>4</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> designs, which are based on a 2-<span><math><msub><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mn>4</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> design and a 2-<span><math><msub><mrow><mo>(</mo><mi>l</mi><mo>,</mo><mn>3</mn><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> design, respectively. Several new infinite classes of simple <em>q</em>-analogs of group divisible designs (<em>q</em>-GDDs) with block dimension 4 are also produced. Making use of the recursive constructions and new <em>q</em>-GDDs, plenty of new infinite series of non-simple subspace 2-designs with block dimension 4 are constructed. We also study the <span><math><mi>GL</mi><mo>(</mo><mi>m</mi><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></math></span>-incidence matrix between 3-subspaces and 4-subspaces. From this, a recursive construction and a new infinite family of non-simple 3-<span><math><msub><mrow><mo>(</mo><mi>m</mi><mi>l</mi><mo>,</mo><mn>4</mn><mo>,</mo><mi>λ</mi><mo>)</mo></mrow><mrow><mi>q</mi></mrow></msub></math></span> designs are produced as well.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"111 ","pages":"Article 102786"},"PeriodicalIF":1.2,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883539","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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