Pub Date : 2026-02-01Epub 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":"2026-02-01","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 : 2026-01-01Epub Date: 2025-07-21DOI: 10.1016/j.ffa.2025.102701
Peter Beelen , Maria Montanucci , Lara Vicino
In this article we continue the work started in [3], explicitly determining the Weierstrass semigroup at any place and the full automorphism group of a known -maximal function field having the third largest genus, for . This function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of -rational places. We show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, is exactly the automorphism group inherited from the Hermitian function field, apart from small values of q.
{"title":"Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 (mod 3)","authors":"Peter Beelen , Maria Montanucci , Lara Vicino","doi":"10.1016/j.ffa.2025.102701","DOIUrl":"10.1016/j.ffa.2025.102701","url":null,"abstract":"<div><div>In this article we continue the work started in <span><span>[3]</span></span>, explicitly determining 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>Y</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> having the third largest genus, for <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>. This function field arises as a Galois subfield of the Hermitian function field, and its uniqueness (with respect to the value of its genus) is a well-known open problem. Knowing the Weierstrass semigroups may provide a key towards solving this problem. Surprisingly enough, <span><math><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> has many different types of Weierstrass semigroups and the set of its Weierstrass places is much richer than its set of <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 show that a similar exceptional behaviour does not occur in terms of automorphisms, that is, <span><math><mrow><mi>Aut</mi></mrow><mo>(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo></math></span> is exactly the automorphism group inherited from the Hermitian function field, apart from small values of <em>q</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102701"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144670374","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 : 2026-01-01Epub Date: 2025-08-12DOI: 10.1016/j.ffa.2025.102717
Sartaj Ul Hasan, Hridesh Kumar
Gutierrez and Urroz (2023) have proposed a family of local permutation polynomials over finite fields of arbitrary characteristic based on a class of symmetric subgroups without fixed points called e-Klenian groups. The polynomials within this family are referred to as e-Klenian polynomials. Furthermore, they have shown the existence of companions for the e-Klenian polynomials when the characteristic of the finite field is odd. Here, we construct three new families of local permutation polynomials over finite fields of even characteristic, and derive a necessary and sufficient condition for each of these families to achieve the maximum possible degree. We also consider the problem of the existence of companions for the e-Klenian polynomials over finite fields of even characteristic. More precisely, we prove that over finite fields of even characteristic, the 0-Klenian polynomials do not have any companions. However, for , we explicitly provide a companion for the e-Klenian polynomials. Moreover, we provide a companion for each of the new families of local permutation polynomials that we introduce.
{"title":"Local permutation polynomials and their companions","authors":"Sartaj Ul Hasan, Hridesh Kumar","doi":"10.1016/j.ffa.2025.102717","DOIUrl":"10.1016/j.ffa.2025.102717","url":null,"abstract":"<div><div>Gutierrez and Urroz (2023) have proposed a family of local permutation polynomials over finite fields of arbitrary characteristic based on a class of symmetric subgroups without fixed points called <em>e</em>-Klenian groups. The polynomials within this family are referred to as <em>e</em>-Klenian polynomials. Furthermore, they have shown the existence of companions for the <em>e</em>-Klenian polynomials when the characteristic of the finite field is odd. Here, we construct three new families of local permutation polynomials over finite fields of even characteristic, and derive a necessary and sufficient condition for each of these families to achieve the maximum possible degree. We also consider the problem of the existence of companions for the <em>e</em>-Klenian polynomials over finite fields of even characteristic. More precisely, we prove that over finite fields of even characteristic, the 0-Klenian polynomials do not have any companions. However, for <span><math><mi>e</mi><mo>≥</mo><mn>1</mn></math></span>, we explicitly provide a companion for the <em>e</em>-Klenian polynomials. Moreover, we provide a companion for each of the new families of local permutation polynomials that we introduce.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102717"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827469","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 : 2026-01-01Epub Date: 2025-08-13DOI: 10.1016/j.ffa.2025.102714
Lior Bary-Soroker, Roy Shmueli
We study a random polynomial of degree n over the finite field , where the coefficients are independent and identically distributed and uniformly chosen from the squares in . Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches as the field size q grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.
{"title":"Irreducibility of polynomials with square coefficients over finite fields","authors":"Lior Bary-Soroker, Roy Shmueli","doi":"10.1016/j.ffa.2025.102714","DOIUrl":"10.1016/j.ffa.2025.102714","url":null,"abstract":"<div><div>We study a random polynomial of degree <em>n</em> over the finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, where the coefficients are independent and identically distributed and uniformly chosen from the squares in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches <span><math><mn>1</mn><mo>/</mo><mi>n</mi><mo>+</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>)</mo></math></span> as the field size <em>q</em> grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102714"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827817","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 : 2026-01-01Epub Date: 2025-07-16DOI: 10.1016/j.ffa.2025.102699
Yunlong Zhu , Chang-An Zhao
For a linear code C over a finite field, if its dual code is equivalent to itself, then the code C is said to be isometry-dual. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes proposed by Han and Ren. Subsequently, two constructions of isometry-dual maximum distance separable (MDS) codes from elliptic curves are presented. The new code length n satisfies when q is even and when q is odd. Additionally, we consider the hull dimension of both constructions. In the case of finite fields with even characteristics, an isometry-dual MDS code is equivalent to a self-dual MDS code and a linear complementary dual MDS code. Finally, we apply our results to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs.
{"title":"On iso-dual MDS codes from elliptic curves","authors":"Yunlong Zhu , Chang-An Zhao","doi":"10.1016/j.ffa.2025.102699","DOIUrl":"10.1016/j.ffa.2025.102699","url":null,"abstract":"<div><div>For a linear code <em>C</em> over a finite field, if its dual code <span><math><msup><mrow><mi>C</mi></mrow><mrow><mo>⊥</mo></mrow></msup></math></span> is equivalent to itself, then the code <em>C</em> is said to be <em>isometry-dual</em>. In this paper, we first confirm a conjecture about the isometry-dual MDS elliptic codes proposed by Han and Ren. Subsequently, two constructions of isometry-dual maximum distance separable (MDS) codes from elliptic curves are presented. The new code length <em>n</em> satisfies <span><math><mi>n</mi><mo>≤</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mo>⌊</mo><mn>2</mn><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>⌋</mo><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> when <em>q</em> is even and <span><math><mi>n</mi><mo>≤</mo><mfrac><mrow><mi>q</mi><mo>+</mo><mo>⌊</mo><mn>2</mn><msqrt><mrow><mi>q</mi></mrow></msqrt><mo>⌋</mo><mo>−</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> when <em>q</em> is odd. Additionally, we consider the hull dimension of both constructions. In the case of finite fields with even characteristics, an isometry-dual MDS code is equivalent to a self-dual MDS code and a linear complementary dual MDS code. Finally, we apply our results to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102699"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144632513","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":"Evaluations of Eisenstein sums","authors":"Ron Evans , Mark Van Veen","doi":"10.1016/j.ffa.2025.102698","DOIUrl":"10.1016/j.ffa.2025.102698","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> be a field of <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> elements. For<span><span><span><math><mrow><mi>k</mi><mo>=</mo><mn>9</mn><mo>,</mo><mn>11</mn><mo>,</mo><mn>13</mn><mo>,</mo><mn>14</mn><mo>,</mo><mn>15</mn><mo>,</mo><mn>16</mn><mo>,</mo><mn>18</mn><mo>,</mo><mn>20</mn><mo>,</mo><mn>22</mn><mo>,</mo><mn>23</mn><mo>,</mo><mn>26</mn><mo>,</mo><mn>30</mn><mo>,</mo><mn>46</mn></mrow></math></span></span></span> we evaluate Eisenstein sums of order <em>k</em> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> when <em>r</em> is the smallest positive integer for which <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>≡</mo><mn>1</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mi>k</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102698"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596554","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 : 2026-01-01Epub Date: 2025-09-04DOI: 10.1016/j.ffa.2025.102720
Rongyin Wang
P. Erdős conjectured in 1962 that on the ring , every set of n congruence classes in that covers the first positive integers also covers the ring . 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 n cosets of ideals in that covers all polynomials whose degree is less than n covers the ring .
P. Erdős于1962年推测，在环Z上，Z上覆盖前2n个正整数的n个同余类的每一个集合也覆盖环Z。这一猜想于1970年由R. B. Crittenden和C. L. Vanden Eynden首次证实。后来，在2019年，P. Balister， B. Bollobás， R. Morris， J. Sahasrabudhe和M. Tiba提供了更透明的证据。本文采用R. B. Crittenden和C. L. Vanden Eynden的方法证明了有限域上多项式环集合中的广义Erdős猜想。我们证明了Fq[x]中覆盖所有阶数小于n的多项式的理想的n个余集的每一个集合覆盖环Fq[x]。
{"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":"2026-01-01","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}
Pub Date : 2026-01-01Epub Date: 2025-07-11DOI: 10.1016/j.ffa.2025.102695
Anna-Maurin Graner
In this paper we present the solution to four 150 year-old closely related problems in the study of polynomials and algebraic codes over a finite field . We give one closed formula each for the factorization of the polynomials for arbitrary , , the n-th cyclotomic polynomial and the composition for arbitrary monic irreducible polynomials , , for arbitrary positive integers n. With a new perspective on these problems we show that the factorization of has a beautiful underlying structure which is completely determined by the order of a in the multiplicative group .
The binomial and the composition were first considered over prime fields by Joseph Alfred Serret in 1866. In recent years, the factorization of has been studied extensively due to the fact that its factors are the generators of the popular constacyclic codes over finite fields. The factorization of and of its famous factor, the n-th cyclotomic polynomial, was first studied over prime fields by Carl Friedrich Gauss (among others) in the middle of the 19th century. Since then, many mathematicians were fascinated by this factorization and nowadays it is needed for the construction of the cyclic codes over finite fields.
Many formulas for the factorization of the four polynomials for special choices of n and a were obtained by a large number of mathematicians. Our formulas naturally extend, cover, combine and complete all of these partial solutions.
本文给出了有限域Fq上多项式和代数码研究中四个有150年历史的密切相关问题的解。对于任意正整数n，对于任意a∈Fq， Xn−1，对于任意单不可约多项式f∈Fq[X]， f≠X，我们分别给出了多项式Xn−a的因式分解和复合f（Xn）的封闭公式。我们从一个新的角度证明了Xn−a的因式分解具有一个美丽的内在结构，它完全由乘法群Fq中a的阶数决定。二项式Xn−a和组成f（Xn）是由Joseph Alfred Serret在1866年首次提出的。近年来，由于Xn−a的因式分解是有限域上常见的常环码的生成因子，因此对Xn−a的因式分解得到了广泛的研究。19世纪中期，卡尔·弗里德里希·高斯（Carl Friedrich Gauss）首先在素数场上研究了Xn−1及其著名的因子n-环多项式的分解。从那时起，许多数学家对这种分解着迷，现在需要它来构造有限域上的循环码。对于n和a的特殊选择，许多数学家得到了四个多项式的因式分解的许多公式。我们的公式自然地扩展、涵盖、组合并完成所有这些部分解。
{"title":"Closed formulas for the generators of all constacyclic codes and for the factorization of Xn − 1, the n-th cyclotomic polynomial and every composition of the form f(Xn) over a finite field for arbitrary positive integers n","authors":"Anna-Maurin Graner","doi":"10.1016/j.ffa.2025.102695","DOIUrl":"10.1016/j.ffa.2025.102695","url":null,"abstract":"<div><div>In this paper we present the solution to four 150 year-old closely related problems in the study of polynomials and algebraic codes over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We give <em>one closed formula each</em> for the factorization of the polynomials <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> for arbitrary <span><math><mi>a</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>, <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span>, the <em>n</em>-th cyclotomic polynomial and the composition <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> for arbitrary monic irreducible polynomials <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span>, <span><math><mi>f</mi><mo>≠</mo><mi>X</mi></math></span>, for <em>arbitrary positive integers n.</em> With a new perspective on these problems we show that the factorization of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> has a beautiful underlying structure which is completely determined by the order of <em>a</em> in the multiplicative group <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>.</div><div>The binomial <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> and the composition <span><math><mi>f</mi><mo>(</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span> were first considered over prime fields by Joseph Alfred Serret in 1866. In recent years, the factorization of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>a</mi></math></span> has been studied extensively due to the fact that its factors are the generators of the popular constacyclic codes over finite fields. The factorization of <span><math><msup><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></math></span> and of its famous factor, the <em>n</em>-th cyclotomic polynomial, was first studied over prime fields by Carl Friedrich Gauss (among others) in the middle of the 19th century. Since then, many mathematicians were fascinated by this factorization and nowadays it is needed for the construction of the cyclic codes over finite fields.</div><div>Many formulas for the factorization of the four polynomials for special choices of <em>n</em> and <em>a</em> were obtained by a large number of mathematicians. Our formulas naturally extend, cover, combine and complete all of these partial solutions.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102695"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596934","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 : 2026-01-01Epub Date: 2025-07-11DOI: 10.1016/j.ffa.2025.102696
Kaloyan Slavov
Let be polynomials in n variables with coefficients in a finite field . We estimate the number of points in such that each value is a nonzero square in . The error term is especially small when the define smooth projective quadrics with nonsingular intersections. We improve the error term in a recent work by Asgarli–Yip on mutual position of smooth quadrics.
{"title":"Square values of several polynomials over a finite field","authors":"Kaloyan Slavov","doi":"10.1016/j.ffa.2025.102696","DOIUrl":"10.1016/j.ffa.2025.102696","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> be polynomials in <em>n</em> variables with coefficients in a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. We estimate the number of points <span><math><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder></math></span> in <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> such that each value <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><munder><mrow><mi>x</mi></mrow><mo>_</mo></munder><mo>)</mo></math></span> is a nonzero square in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>. The error term is especially small when the <span><math><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> define smooth projective quadrics with nonsingular intersections. We improve the error term in a recent work by Asgarli–Yip on mutual position of smooth quadrics.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102696"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596553","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 : 2026-01-01Epub Date: 2025-07-07DOI: 10.1016/j.ffa.2025.102686
Rajendra Prasad Rajpurohit, Maheshanand Bhaintwal
In this paper, we present a novel construction of maximally recoverable codes with two-level hierarchical locality using a parity-check matrix approach. The construction given in this paper utilizes Gabidulin codes for mid-level heavy parities and linearized Reed-Solomon codes for global heavy parities. When the number of local sets is small, this construction performs better than the previously known constructions as the field size required in our construction is smaller for such cases, making it useful for practical scenarios in distributed data storage systems. We also consider a special case of our construction when the number of global parities is fixed and is equal to 1. In this case, our construction performs better when the number of local sets is small and the number of mid-level parities is even.
{"title":"A new construction of maximally recoverable codes with hierarchical locality","authors":"Rajendra Prasad Rajpurohit, Maheshanand Bhaintwal","doi":"10.1016/j.ffa.2025.102686","DOIUrl":"10.1016/j.ffa.2025.102686","url":null,"abstract":"<div><div>In this paper, we present a novel construction of maximally recoverable codes with two-level hierarchical locality using a parity-check matrix approach. The construction given in this paper utilizes Gabidulin codes for mid-level heavy parities and linearized Reed-Solomon codes for global heavy parities. When the number of local sets is small, this construction performs better than the previously known constructions as the field size required in our construction is smaller for such cases, making it useful for practical scenarios in distributed data storage systems. We also consider a special case of our construction when the number of global parities is fixed and is equal to 1. In this case, our construction performs better when the number of local sets is small and the number of mid-level parities is even.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102686"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144570348","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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