Finite Fields and Their Applications最新文献

{"title":"Brocard-Ramanujan problem for polynomials over finite fields","authors":"Wataru Takeda","doi":"10.1016/j.ffa.2025.102731","DOIUrl":"10.1016/j.ffa.2025.102731","url":null,"abstract":"<div><div>The Brocard-Ramanujan problem is an unsolved number theory problem to find integer solutions <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><mi>n</mi><mo>!</mo></math></span>. In this paper, we consider this problem over polynomial rings <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>T</mi><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is a finite field with <em>q</em> elements. We find all solutions to the equation <span><math><msup><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><mn>1</mn><mo>=</mo><msub><mrow><mi>Π</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>Π</mi></mrow><mrow><mi>C</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denotes the Carlitz factorial. More precisely, we characterize all solutions and prove that there are infinitely many solutions if and only if <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> is an extension of <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. This characterization is achieved without using the Mason-Stothers theorem, analogous to the abc conjecture for integers.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102731"},"PeriodicalIF":1.2,"publicationDate":"2025-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221461","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":"Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings","authors":"Sanjit Bhowmick ,&nbsp;Kuntal Deka ,&nbsp;Alexandre Fotue Tabue ,&nbsp;Edgar Martínez-Moro","doi":"10.1016/j.ffa.2025.102732","DOIUrl":"10.1016/j.ffa.2025.102732","url":null,"abstract":"<div><div>This paper investigates the algebraic structure of complementary pairs of additive cyclic codes over a finite commutative chain ring of odd characteristic. We demonstrate that for every additive complementary pair of additive codes, both constituent codes are free modules. Moreover, we present a necessary and sufficient condition for a pair of additive codes over a finite commutative chain ring of odd characteristic to form an additive complementary pair. Finally, we show that, in the case of a complementary pair of additive cyclic codes over a finite chain ring of odd characteristic, one of the codes is permutation equivalent to the trace dual of the other.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102732"},"PeriodicalIF":1.2,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221462","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the study of semi-involutory and semi-orthogonal matrices","authors":"Yogesh Kumar ,&nbsp;Susanta Samanta ,&nbsp;P.R. Mishra ,&nbsp;Atul Gaur","doi":"10.1016/j.ffa.2025.102730","DOIUrl":"10.1016/j.ffa.2025.102730","url":null,"abstract":"<div><div>This paper investigates semi-involutory and semi-orthogonal matrices, presenting two algorithms for verifying these properties for <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. The algorithms significantly reduce computational complexity by avoiding the need for non-singular diagonal matrices. The structure of circulant matrices with these properties is also explored, including the derivation of the exact form of their corresponding diagonal matrices. We further investigate MDS matrices with semi-involutory and semi-orthogonal properties. We prove that semi-involutory circulant matrices of order <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, cannot be MDS matrices. Additionally, we show that semi-orthogonal circulant matrices of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span> cannot be MDS. Finally, explicit formulas for counting <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> semi-involutory and semi-orthogonal MDS matrices over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span> are derived, and exact counts for <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> semi-involutory and semi-orthogonal MDS matrices over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span> for <span><math><mi>m</mi><mo>=</mo><mn>3</mn></math></span> and 4 are provided.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102730"},"PeriodicalIF":1.2,"publicationDate":"2025-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221460","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":"Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q≡0(mod3)","authors":"Peter Beelen ,&nbsp;Maria Montanucci ,&nbsp;Lara Vicino","doi":"10.1016/j.ffa.2025.102729","DOIUrl":"10.1016/j.ffa.2025.102729","url":null,"abstract":"<div><div>In this article, we explicitly determine the Weierstrass semigroup at any place and the full automorphism group of a known <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-maximal function field <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, which is realised as a Galois subfield of the Hermitian function field and has the third largest genus, for <span><math><mi>q</mi><mo>≡</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>. This completes the work contained in <span><span>[3]</span></span> and <span><span>[4]</span></span>, where the cases <span><math><mi>q</mi><mo>≡</mo><mn>2</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span> and <span><math><mi>q</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>, respectively, were studied. Like for these other two cases, the problem of determining the uniqueness of the function field <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>, with respect to the value of its genus, is still open. The knowledge of the Weierstrass semigroups may be instrumental in finding a solution to this problem, as it happened to be the case for the function fields with the largest <span><span>[11]</span></span> and second largest genera <span><span>[1]</span></span>, <span><span>[7]</span></span>. Similarly to what observed in <span><span>[3]</span></span> and <span><span>[4]</span></span>, also in the case of <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> we find that many different types of Weierstrass semigroups appear, and that the set of Weierstrass places contains also non-<span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math></span>-rational places. We also determine <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span>, which turns out to be exactly the automorphism group inherited from the Hermitian function field, apart from the case <span><math><mi>q</mi><mo>=</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102729"},"PeriodicalIF":1.2,"publicationDate":"2025-09-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145110037","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 dual codes of two families of BCH codes","authors":"Haojie Xu ,&nbsp;Xia Wu ,&nbsp;Wei Lu ,&nbsp;Xiwang Cao","doi":"10.1016/j.ffa.2025.102721","DOIUrl":"10.1016/j.ffa.2025.102721","url":null,"abstract":"<div><div>In this paper, we present an infinite family of MDS codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span> and two infinite families of almost MDS codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span> for any prime <em>p</em>, by investigating the parameters of the dual codes of two families of BCH codes. Notably, these almost MDS codes include two infinite families of near MDS codes over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>3</mn></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span>, resolving a conjecture posed by Geng et al. in 2022. Furthermore, we demonstrate that both of these almost MDS codes and their dual codes hold infinite families of 3-designs over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span> for any prime <em>p</em>. Additionally, we study the subfield subcodes of these families of MDS and near MDS codes, and provide several binary, ternary, and quaternary codes with best known parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102721"},"PeriodicalIF":1.2,"publicationDate":"2025-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097424","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":"Mean value theorems for short rational exponential sums","authors":"Doowon Koh ,&nbsp;Igor E. Shparlinski","doi":"10.1016/j.ffa.2025.102719","DOIUrl":"10.1016/j.ffa.2025.102719","url":null,"abstract":"<div><div>We obtain finite field analogues of a series of recent results on various mean value theorems for Weyl sums. Instead of the Vinogradov Mean Value Theorem, our results rest on the classical argument of Mordell, combined with several other ideas.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102719"},"PeriodicalIF":1.2,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145020639","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":"Large cyclic subspace codes over finite fields","authors":"He Zhang ,&nbsp;Chunming Tang ,&nbsp;Xiwang Cao ,&nbsp;Gaojun Luo","doi":"10.1016/j.ffa.2025.102722","DOIUrl":"10.1016/j.ffa.2025.102722","url":null,"abstract":"<div><div>Cyclic subspace codes play a crucial role in random network coding. Designing such cyclic subspace codes with the largest possible code size and minimum distance remains a classical problem. Roth et al. (2018) <span><span>[28]</span></span> first investigated optimal cyclic subspace codes via Sidon spaces and proved that the orbit of a Sidon space is an optimal cyclic subspace code with full-length orbit. This paper introduces a new method, namely the intermediate extension field, to construct Sidon spaces and cyclic subspace codes. The main results show that our new codes over intermediate fields have optimal minimum distance and contain more codewords than known constructions. Therefore, this work improves the lower bound of optimal cyclic subspace codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102722"},"PeriodicalIF":1.2,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145011304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On an Erdős-type conjecture on Fq[x]","authors":"Rongyin Wang","doi":"10.1016/j.ffa.2025.102720","DOIUrl":"10.1016/j.ffa.2025.102720","url":null,"abstract":"<div><div>P. Erdős conjectured in 1962 that on the ring <span><math><mi>Z</mi></math></span>, every set of <em>n</em> congruence classes in <span><math><mi>Z</mi></math></span> that covers the first <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> positive integers also covers the ring <span><math><mi>Z</mi></math></span>. This conjecture was first confirmed in 1970 by R. B. Crittenden and C. L. Vanden Eynden. Later, in 2019, P. Balister, B. Bollobás, R. Morris, J. Sahasrabudhe, and M. Tiba provided a more transparent proof. In this paper, we follow the approach used by R. B. Crittenden and C. L. Vanden Eynden to prove the generalized Erdős' conjecture in the setting of polynomial rings over finite fields. We prove that every set of <em>n</em> cosets of ideals in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> that covers all polynomials whose degree is less than <em>n</em> covers the ring <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102720"},"PeriodicalIF":1.2,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144988706","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 MDS Euclidean self-dual codes via multiple subsets","authors":"Weirong Meng ,&nbsp;Weijun Fang ,&nbsp;Fang-Wei Fu ,&nbsp;Haiyan Zhou ,&nbsp;Ziyi Gu","doi":"10.1016/j.ffa.2025.102718","DOIUrl":"10.1016/j.ffa.2025.102718","url":null,"abstract":"<div><div>MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85%, which is much higher than previously reported results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102718"},"PeriodicalIF":1.2,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144912895","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 distribution of a-numbers of hyperelliptic curves in characteristic three","authors":"Derek Garton ,&nbsp;Jeffrey Lin Thunder ,&nbsp;Colin Weir","doi":"10.1016/j.ffa.2025.102715","DOIUrl":"10.1016/j.ffa.2025.102715","url":null,"abstract":"<div><div>In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus <em>g</em> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with a given <em>a</em>-number. In characteristic three this method gives exact probabilities for curves of the form <span><math><msup><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span> monic and cubefree—probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of <em>q</em>) for these probabilities when restricting to squarefree <em>f</em>. As a consequence, for positive integers <em>a</em> and <em>g</em> we show that the nonempty strata of the moduli space of hyperelliptic curves of genus <em>g</em> consisting of those curves with <em>a</em>-number <em>a</em> are of codimension <span><math><mn>2</mn><mi>a</mi><mo>−</mo><mn>1</mn></math></span>. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are <span><math><mi>a</mi><mo>(</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.—one that matches the available data.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102715"},"PeriodicalIF":1.2,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879454","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}