Pub Date : 2025-10-17DOI: 10.1016/j.ffa.2025.102741
Sujata Bansal, Pramod Kumar Kewat
This work explores binary polycyclic codes associated with the polynomial , which is the -th power of for every integer . We provide an in-depth structural analysis of these codes and compute the exact Hamming distance of each of these binary polycyclic codes. Furthermore, we determine the parity-check matrices and examine the Euclidean duals and annihilator duals for these polycyclic codes. Our analysis reveals that these codes are reversible and, in certain cases, are Linear Complementary Dual (LCD) codes. This discovery highlights the potential of these codes in practical applications such as communication systems, data storage, consumer electronics, and cryptography. We also propose a conjecture that suggests all such polycyclic codes can be LCD.
{"title":"Binary polycyclic codes associated with x2η+1+x2η+1: Hamming distance, duality, reversibility and LCD properties","authors":"Sujata Bansal, Pramod Kumar Kewat","doi":"10.1016/j.ffa.2025.102741","DOIUrl":"10.1016/j.ffa.2025.102741","url":null,"abstract":"<div><div>This work explores binary polycyclic codes associated with the polynomial <span><math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>η</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>η</mi></mrow></msup></mrow></msup><mo>+</mo><mn>1</mn></math></span>, which is the <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>η</mi></mrow></msup></math></span>-th power of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>x</mi><mo>+</mo><mn>1</mn></math></span> for every integer <span><math><mi>η</mi><mo>≥</mo><mn>1</mn></math></span>. We provide an in-depth structural analysis of these codes and compute the exact Hamming distance of each of these binary polycyclic codes. Furthermore, we determine the parity-check matrices and examine the Euclidean duals and annihilator duals for these polycyclic codes. Our analysis reveals that these codes are reversible and, in certain cases, are Linear Complementary Dual (LCD) codes. This discovery highlights the potential of these codes in practical applications such as communication systems, data storage, consumer electronics, and cryptography. We also propose a conjecture that suggests all such polycyclic codes can be LCD.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102741"},"PeriodicalIF":1.2,"publicationDate":"2025-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321166","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 : 2025-10-16DOI: 10.1016/j.ffa.2025.102740
Xue Jia , Fengwei Li , Huan Sun , Qin Yue
In this paper, we investigate polynomials of the form , where , p is a prime, and k divides n. By introducing a new approach based on the projective general linear group, we show that the number of zeros of in belongs to , and provide explicit criteria on b for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where has at least one zero, we determine its complete irreducible factorization over .
{"title":"Irreducible factorizations of polynomials xpk+1−bx+b over a finite field","authors":"Xue Jia , Fengwei Li , Huan Sun , Qin Yue","doi":"10.1016/j.ffa.2025.102740","DOIUrl":"10.1016/j.ffa.2025.102740","url":null,"abstract":"<div><div>In this paper, we investigate polynomials of the form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi></math></span>, where <span><math><mn>0</mn><mo>≠</mo><mi>b</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, <em>p</em> is a prime, and <em>k</em> divides <em>n</em>. By introducing a new approach based on the projective general linear group, we show that the number of zeros of <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> belongs to <span><math><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>}</mo></math></span>, and provide explicit criteria on <em>b</em> for each case. We also count the number of such polynomials corresponding to each possible number of zeros. Moreover, for the cases where <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> has at least one zero, we determine its complete irreducible factorization over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102740"},"PeriodicalIF":1.2,"publicationDate":"2025-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321165","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 : 2025-10-10DOI: 10.1016/j.ffa.2025.102736
Dušan Dragutinović
We construct a family of smooth supersingular curves of genus 5 in characteristic 2 with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus 5, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-2 curve. We also provide an explicit parametrization of this family.
{"title":"An unusual family of supersingular curves of genus five in characteristic two","authors":"Dušan Dragutinović","doi":"10.1016/j.ffa.2025.102736","DOIUrl":"10.1016/j.ffa.2025.102736","url":null,"abstract":"<div><div>We construct a family of smooth supersingular curves of genus 5 in characteristic 2 with several notable features: its dimension matches the expected dimension of any component of the supersingular locus in genus 5, its members are non-hyperelliptic curves with non-trivial automorphism groups, and each curve in the family admits a double cover structure over both an elliptic curve and a genus-2 curve. We also provide an explicit parametrization of this family.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102736"},"PeriodicalIF":1.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267945","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 : 2025-10-10DOI: 10.1016/j.ffa.2025.102734
Chin Hei Chan , Zhiguo Ding , Nian Li , Xi Xie , Maosheng Xiong , Michael E. Zieve
Let , where is the finite field of order and for some positive integer m. Tu et al. (Finite Fields Appl. 68: 1-20, 2020) proposed a sufficient condition under which is a complete permutation on . In this paper, we show that this sufficient condition is also necessary, and when is a complete permutation, then and are simultaneously linear equivalent to and for some satisfying . This result leads to a complete characterization of the complete permutation quadrinomials of the above form .
{"title":"On a class of complete permutation quadrinomials","authors":"Chin Hei Chan , Zhiguo Ding , Nian Li , Xi Xie , Maosheng Xiong , Michael E. Zieve","doi":"10.1016/j.ffa.2025.102734","DOIUrl":"10.1016/j.ffa.2025.102734","url":null,"abstract":"<div><div>Let <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn><mi>q</mi></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn><mi>q</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>q</mi><mo>+</mo><mn>2</mn></mrow></msup><mo>+</mo><mi>d</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><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><mi>x</mi><mo>]</mo></math></span>, where <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> is the finite field of order <span><math><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span> for some positive integer <em>m</em>. Tu et al. (Finite Fields Appl. 68: 1-20, 2020) proposed a sufficient condition under which <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a complete permutation on <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>. In this paper, we show that this sufficient condition is also necessary, and when <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a complete permutation, then <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> and <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>x</mi></math></span> are simultaneously linear equivalent to <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mover><mrow><mi>x</mi></mrow><mo>‾</mo></mover></math></span> and <span><math><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mover><mrow><mi>x</mi></mrow><mo>‾</mo></mover><mo>+</mo><mi>γ</mi><mi>x</mi></math></span> for some <span><math><mi>γ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> satisfying <span><math><mrow><mi>ord</mi></mrow><mo>(</mo><msup><mrow><mi>γ</mi></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><mo>=</mo><mn>3</mn></math></span>. This result leads to a complete characterization of the complete permutation quadrinomials of the above form <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102734"},"PeriodicalIF":1.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267944","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 : 2025-10-10DOI: 10.1016/j.ffa.2025.102738
Shushi Harashita, Yuya Yamamoto
Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curves in Legendre form is separable. In this paper, we get an analogous result for curves of genus 2 in Rosenhain form. More precisely we show that the ideal determining the superspeciality of the curve has multiplicity one at every superspecial point. Igusa used a Picard-Fucks differential operator annihilating a Gauß hypergeometric series. We shall use the Lauricella system (of type D) of hypergeometric differential equations in three variables.
{"title":"The multiplicity-one theorem for the superspeciality of curves of genus two","authors":"Shushi Harashita, Yuya Yamamoto","doi":"10.1016/j.ffa.2025.102738","DOIUrl":"10.1016/j.ffa.2025.102738","url":null,"abstract":"<div><div>Igusa proved in 1958 that the polynomial determining the supersingularity of elliptic curves in Legendre form is separable. In this paper, we get an analogous result for curves of genus 2 in Rosenhain form. More precisely we show that the ideal determining the superspeciality of the curve has multiplicity one at every superspecial point. Igusa used a Picard-Fucks differential operator annihilating a Gauß hypergeometric series. We shall use the Lauricella system (of type D) of hypergeometric differential equations in three variables.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102738"},"PeriodicalIF":1.2,"publicationDate":"2025-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267943","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 : 2025-10-08DOI: 10.1016/j.ffa.2025.102735
Thomas Karam
We identify a new sufficient condition on linear forms which guarantees that every subset of on which none of has full image has a density which tends to 0 with k. The condition is much weaker than the condition usually used to guarantee that takes each value of with probability close to when x is chosen uniformly at random in the Boolean cube . The density is at most quasipolynomially small in k, a bound that is necessarily close to sharp.
{"title":"On small densities defined without pseudorandomness","authors":"Thomas Karam","doi":"10.1016/j.ffa.2025.102735","DOIUrl":"10.1016/j.ffa.2025.102735","url":null,"abstract":"<div><div>We identify a new sufficient condition on linear forms <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>:</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msubsup><mo>→</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span> which guarantees that every subset of <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span> on which none of <span><math><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> has full image has a density which tends to 0 with <em>k</em>. The condition is much weaker than the condition usually used to guarantee that <span><math><mo>(</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>ϕ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> takes each value of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> with probability close to <span><math><msup><mrow><mi>p</mi></mrow><mrow><mo>−</mo><mi>k</mi></mrow></msup></math></span> when <em>x</em> is chosen uniformly at random in the Boolean cube <span><math><msup><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow><mrow><mi>n</mi></mrow></msup></math></span>. The density is at most quasipolynomially small in <em>k</em>, a bound that is necessarily close to sharp.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102735"},"PeriodicalIF":1.2,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267946","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}
Let N denote the number of solutions to the generalized Markoff-Hurwitz-type equation over the finite field , where are positive integers, and for , with and . Using techniques from algebraic geometry, we provide an estimate for N and establish conditions under which the equation admits solutions where all are nonzero.
{"title":"Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields","authors":"Miriam Abdón , Daniela Alves de Oliveira , Juliane Capaverde , Mariana Pérez , Melina Privitelli","doi":"10.1016/j.ffa.2025.102733","DOIUrl":"10.1016/j.ffa.2025.102733","url":null,"abstract":"<div><div>Let <em>N</em> denote the number of solutions to the generalized Markoff-Hurwitz-type equation<span><span><span><math><msup><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msubsup><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub><msubsup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow><mrow><mi>m</mi></mrow></msubsup><mo>+</mo><mi>a</mi><mo>)</mo></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><mi>b</mi><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span></span></span> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where <span><math><mi>m</mi><mo>,</mo><mi>k</mi></math></span> are positive integers, and <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span> for <span><math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>, with <span><math><mi>k</mi><mo>,</mo><mi>m</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. Using techniques from algebraic geometry, we provide an estimate for <em>N</em> and establish conditions under which the equation admits solutions where all <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> are nonzero.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102733"},"PeriodicalIF":1.2,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267942","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 : 2025-10-07DOI: 10.1016/j.ffa.2025.102737
Bo-Hae Im , Hansol Kim
For a field K of characteristic , let be an elliptic curve defined over the function field in two variables s and t. For a non-negative positive integer e and a positive integer N which is not divisible by p, we prove that if , then the automorphism group of the normal extension over is isomorphic to . Applying this result, we also determine the automorphism group of the normal extension for a general field K of characteristic .
{"title":"Automorphism groups of the fields of definition of torsion points of elliptic curves in characteristic ≥5","authors":"Bo-Hae Im , Hansol Kim","doi":"10.1016/j.ffa.2025.102737","DOIUrl":"10.1016/j.ffa.2025.102737","url":null,"abstract":"<div><div>For a field <em>K</em> of characteristic <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span>, let <span><math><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mi>s</mi><mi>x</mi><mo>+</mo><mi>t</mi></math></span> be an elliptic curve defined over the function field <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> in two variables <em>s</em> and <em>t</em>. For a non-negative positive integer <em>e</em> and a positive integer <em>N</em> which is not divisible by <em>p</em>, we prove that if <span><math><mi>K</mi><mo>⊇</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow><mrow><mi>alg</mi></mrow></msubsup></math></span>, then the automorphism group of the normal extension <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mi>N</mi><mo>]</mo></mrow><mo>)</mo></mrow></math></span> over <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></math></span> is isomorphic to <span><math><msup><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mi>Z</mi><mo>)</mo></mrow><mrow><mo>×</mo></mrow></msup><mo>×</mo><msub><mrow><mi>SL</mi></mrow><mrow><mn>2</mn></mrow></msub><mspace></mspace><mrow><mo>(</mo><mi>Z</mi><mo>/</mo><mi>N</mi><mi>Z</mi><mo>)</mo></mrow></math></span>. Applying this result, we also determine the automorphism group of the normal extension <span><math><mi>K</mi><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mo>(</mo><msub><mrow><mi>E</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><mrow><mo>[</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup><mi>N</mi><mo>]</mo></mrow><mo>)</mo></mrow></math></span> for a general field <em>K</em> of characteristic <span><math><mi>p</mi><mo>≥</mo><mn>5</mn></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102737"},"PeriodicalIF":1.2,"publicationDate":"2025-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145267941","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 : 2025-09-30DOI: 10.1016/j.ffa.2025.102731
Wataru Takeda
The Brocard-Ramanujan problem is an unsolved number theory problem to find integer solutions to . In this paper, we consider this problem over polynomial rings , where is a finite field with q elements. We find all solutions to the equation , where denotes the Carlitz factorial. More precisely, we characterize all solutions and prove that there are infinitely many solutions if and only if is an extension of . This characterization is achieved without using the Mason-Stothers theorem, analogous to the abc conjecture for integers.
{"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}
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.
{"title":"Trace duality and additive complementary pairs of additive cyclic codes over finite chain rings","authors":"Sanjit Bhowmick , Kuntal Deka , Alexandre Fotue Tabue , 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}