{"title":"An elementary proof of Bridy's theorem","authors":"Eric Rowland , Manon Stipulanti , Reem Yassawi","doi":"10.1016/j.ffa.2025.102621","DOIUrl":"10.1016/j.ffa.2025.102621","url":null,"abstract":"<div><div>Christol's theorem states that a power series with coefficients in a finite field is algebraic if and only if its coefficient sequence is automatic. A natural question is how the size of a polynomial describing such a sequence relates to the size of an automaton describing the same sequence. Bridy used tools from algebraic geometry to bound the size of the minimal automaton for a sequence, given its minimal polynomial. We produce a new proof of Bridy's bound by embedding algebraic sequences as diagonals of rational functions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102621"},"PeriodicalIF":1.2,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"BRK-type sets over finite fields","authors":"Charlotte Trainor","doi":"10.1016/j.ffa.2025.102624","DOIUrl":"10.1016/j.ffa.2025.102624","url":null,"abstract":"<div><div>A Besicovitch-Rado-Kinney (BRK) set in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is a Borel set that contains a <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional sphere of radius <em>r</em>, for each <span><math><mi>r</mi><mo>></mo><mn>0</mn></math></span>. It is known that such sets have Hausdorff dimension <em>n</em> from the work of Kolasa and Wolff. In this paper, we consider an analogous problem over a finite field, <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We define BRK-type sets in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, and establish lower bounds on the size of such sets using techniques introduced by Dvir's proof of the finite field Kakeya conjecture.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102624"},"PeriodicalIF":1.2,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696630","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":"Non-isomorphic maximal function fields of genus q − 1","authors":"Jonathan Niemann","doi":"10.1016/j.ffa.2025.102618","DOIUrl":"10.1016/j.ffa.2025.102618","url":null,"abstract":"<div><div>The classification of maximal function fields over a finite field is a difficult open problem, and even determining isomorphism classes among known function fields is challenging in general. We study a particular family of maximal function fields defined over a finite field with <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> elements, where <em>q</em> is the power of an odd prime. When <span><math><mi>d</mi><mo>:</mo><mo>=</mo><mo>(</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span> is a prime, this family is known to contain a large number of non-isomorphic function fields of the same genus and with the same automorphism group. We compute the automorphism group and isomorphism classes also in the case where <em>d</em> is not a prime.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102618"},"PeriodicalIF":1.2,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143684818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Characterizations of NMDS codes and a proof of the Geng-Yang-Zhang-Zhou conjecture","authors":"Shiyuan Qiang, Huakai Wei, Shaofang Hong","doi":"10.1016/j.ffa.2025.102616","DOIUrl":"10.1016/j.ffa.2025.102616","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be the finite field of <em>q</em> elements, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> with <em>p</em> being a prime number and <em>m</em> being a positive integer. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>δ</mi><mo>,</mo><mi>h</mi><mo>)</mo></mrow></msub></math></span> be a class of BCH codes of length <em>n</em> and designed distance <em>δ</em>. A linear code <span><math><mi>C</mi></math></span> is said to be maximum distance separable (MDS) if the minimum distance <span><math><mi>d</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. If <span><math><mi>d</mi><mo>=</mo><mi>n</mi><mo>−</mo><mi>k</mi></math></span>, then <span><math><mi>C</mi></math></span> is called an almost MDS (AMDS) code. Moreover, if both of <span><math><mi>C</mi></math></span> and its dual code <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> are AMDS, then <span><math><mi>C</mi></math></span> is called a near MDS (NMDS) code. In <span><span>[9]</span></span>, Geng, Yang, Zhang and Zhou proved that the BCH code <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></msub></math></span> is an almost MDS code, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>3</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span> and <em>m</em> is an odd integer, and they also showed that its parameters is <span><math><mo>[</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>]</mo></math></span>. Furthermore, they proposed a conjecture stating that the dual code <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mrow><mo>⊥</mo></mrow></msubsup></math></span> is also an AMDS code with parameters <span><math><mo>[</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>q</mi><mo>−</mo><mn>3</mn><mo>]</mo></math></span>. In this paper, we introduce the concept of subset code and use it together with the MacWilliams identity to establish characterizations for the dual code of an AMDS code to be an AMDS code. Then by this criteria, we show that the Geng-Yang-Zhang-Zhou conjecture is true. Our result together with the Geng-Yang-Zhang-Zhou theorem implies that the BCH code <span><math><msub><mrow><mi>C</mi></mrow><mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>q</mi><mo>+</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow></msub></math></span> is an NMDS code.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102616"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143619240","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":"Central and non-central metacyclic codes","authors":"Seema Chahal, Sugandha Maheshwary","doi":"10.1016/j.ffa.2025.102615","DOIUrl":"10.1016/j.ffa.2025.102615","url":null,"abstract":"<div><div>In this article, the primitive central idempotents of finite semisimple group algebra of split metacyclic groups are explicitly written and their supports are evaluated. This is used to derive substantial information on the distance of the corresponding central codes. The dimensions and bases for these codes are also obtained. Some central and non-central good codes are produced.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102615"},"PeriodicalIF":1.2,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143620350","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}
Leandro Cagliero , Allen Herman , Fernando Szechtman
{"title":"Artin-Schreier towers of finite fields","authors":"Leandro Cagliero , Allen Herman , Fernando Szechtman","doi":"10.1016/j.ffa.2025.102606","DOIUrl":"10.1016/j.ffa.2025.102606","url":null,"abstract":"<div><div>Given a prime number <em>p</em>, we consider the tower of finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>⊂</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><mo>⋯</mo></math></span>, where each step corresponds to an Artin-Schreier extension of degree <em>p</em>, so that for <span><math><mi>i</mi><mo>≥</mo><mn>0</mn></math></span>, <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>[</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></math></span>, where <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a root of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>−</mo><mi>X</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><msup><mrow><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with <span><math><msub><mrow><mi>c</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>. We extend and strengthen to arbitrary primes prior work of Popovych for <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> on the multiplicative order <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> of the given generator <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> over <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>. In particular, for <span><math><mi>i</mi><mo>≥</mo><mn>0</mn></math></span>, we show that <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>O</mi><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span>, except only when <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and <span><math><mi>i</mi><mo>=</mo><mn>1</mn></math></span>, and that <span><math><mi>O</mi><mo>(</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span> is equal to the product of the orders of <span><math><msub><mrow><mi>c</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> modulo <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>j</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>×</mo></mrow></m","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"106 ","pages":"Article 102606"},"PeriodicalIF":1.2,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143611110","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 existence of Galois self-dual GRS and TGRS codes","authors":"Shixin Zhu, Ruhao Wan","doi":"10.1016/j.ffa.2025.102608","DOIUrl":"10.1016/j.ffa.2025.102608","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> be a prime power and <em>e</em> be an integer with <span><math><mn>0</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>. <em>e</em>-Galois self-dual codes are generalizations of Euclidean <span><math><mo>(</mo><mi>e</mi><mo>=</mo><mn>0</mn><mo>)</mo></math></span> and Hermitian (<span><math><mi>e</mi><mo>=</mo><mfrac><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> with even <em>m</em>) self-dual codes. In this paper, for a linear code <span><math><mi>C</mi></math></span> and a nonzero vector <span><math><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>, we give a sufficient and necessary condition for the dual extended code <span><math><munder><mrow><mi>C</mi></mrow><mo>_</mo></munder><mo>[</mo><mi>u</mi><mo>]</mo></math></span> of <span><math><mi>C</mi></math></span> to be <em>e</em>-Galois self-orthogonal. From this, a new systematic approach is proposed to prove the existence of <em>e</em>-Galois self-dual codes. By this method, we prove that <em>e</em>-Galois self-dual (extended) generalized Reed-Solomon (GRS) codes of length <span><math><mi>n</mi><mo>></mo><mi>min</mi><mo></mo><mo>{</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>−</mo><mi>e</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>}</mo></math></span> do not exist, where <span><math><mn>1</mn><mo>≤</mo><mi>e</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></math></span>. Moreover, based on the non-GRS properties of twisted GRS (TGRS) codes, we show that in many cases <em>e</em>-Galois self-dual (extended) TGRS codes do not exist. Furthermore, we present a sufficient and necessary condition for <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span>-TGRS codes to be Hermitian self-dual, and then construct several new classes of Hermitian self-dual <span><math><mo>(</mo><mo>+</mo><mo>)</mo></math></span>-TGRS and <span><math><mo>(</mo><mo>⁎</mo><mo>)</mo></math></span>-TGRS codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102608"},"PeriodicalIF":1.2,"publicationDate":"2025-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143562784","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":"Polynomial p-adic low-discrepancy sequences","authors":"Christian Weiß","doi":"10.1016/j.ffa.2025.102607","DOIUrl":"10.1016/j.ffa.2025.102607","url":null,"abstract":"<div><div>The classic example of a low-discrepancy sequence in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> is <span><math><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>=</mo><mi>a</mi><mi>n</mi><mo>+</mo><mi>b</mi></math></span> with <span><math><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow><mrow><mo>×</mo></mrow></msubsup></math></span> and <span><math><mi>b</mi><mo>∈</mo><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>. Here we address the non-linear case and show that a polynomial <em>f</em> generates a low-discrepancy sequence in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> if and only if it is a permutation polynomial <span><math><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. By this it is possible to construct non-linear examples of low-discrepancy sequences in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> for all primes <em>p</em>. Moreover, we prove a criterion which decides for any given polynomial in <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> with <span><math><mi>p</mi><mo>∈</mo><mrow><mo>{</mo><mn>3</mn><mo>,</mo><mn>5</mn><mo>,</mo><mn>7</mn><mo>}</mo></mrow></math></span> if it generates a low-discrepancy sequence. We also discuss connections to the theories of Poissonian pair correlations and real discrepancy.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102607"},"PeriodicalIF":1.2,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143552862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Minjia Shi , Xinpeng Bian , Ferruh Özbudak , Patrick Solé
{"title":"Bound on the minimum distance of double circulant cubic residue codes","authors":"Minjia Shi , Xinpeng Bian , Ferruh Özbudak , Patrick Solé","doi":"10.1016/j.ffa.2025.102605","DOIUrl":"10.1016/j.ffa.2025.102605","url":null,"abstract":"<div><div>We study a class of pure double circulant binary codes attached to the cyclotomy of order 3 with respect to a prime <span><math><mi>p</mi><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>3</mn><mo>)</mo></math></span>. The minimum distance is bounded below by an argument involving cyclotomic numbers and Weil inequality for multiplicative character sums.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102605"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143489010","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}
Sophie Huczynska , Laura Johnson , Maura B. Paterson
{"title":"Beyond uniform cyclotomy","authors":"Sophie Huczynska , Laura Johnson , Maura B. Paterson","doi":"10.1016/j.ffa.2025.102604","DOIUrl":"10.1016/j.ffa.2025.102604","url":null,"abstract":"<div><div>Cyclotomy, the study of cyclotomic classes and cyclotomic numbers, is an area of number theory first studied by Gauss. It has natural applications in discrete mathematics and information theory. Despite this long history, there are significant limitations to what is known explicitly about cyclotomic numbers, which limits the use of cyclotomy in applications. The main explicit tool available is that of uniform cyclotomy, introduced by Baumert, Mills and Ward in 1982. In this paper, we present an extension of uniform cyclotomy which gives a direct method for evaluating all cyclotomic numbers over <span><math><mrow><mi>GF</mi></mrow><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> of order dividing <span><math><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>n</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>, for any prime power <em>q</em> and <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, which does not use character theory nor direct calculation in the field. This allows the straightforward evaluation of many cyclotomic numbers for which other methods are unknown or impractical, extending the currently limited portfolio of tools to work with cyclotomic numbers. Our methods exploit connections between cyclotomy, Singer difference sets and finite geometry.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"105 ","pages":"Article 102604"},"PeriodicalIF":1.2,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143488563","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号
