Pub Date : 2025-10-29DOI: 10.1016/j.ffa.2025.102750
Junfeng Jia, Yanxun Chang
Flag codes, as a generalization of subspace codes, can transmit more information since the subspace channel is used many times. In this paper, we construct optimum distance flag codes of the (generalized) full admissible type on with cardinality , where with and . Let denote the maximum cardinality of such codes. We provide a lower bound for this quantity. We further present a systematic construction of cardinality-consistent flag codes with larger cardinality for general flag distances. By the composition of subspace polynomials, we construct cardinality-consistent cyclic flag codes on with larger cardinality than those presented in the literature.
{"title":"Cardinality-consistent flag codes with larger cardinality","authors":"Junfeng Jia, Yanxun Chang","doi":"10.1016/j.ffa.2025.102750","DOIUrl":"10.1016/j.ffa.2025.102750","url":null,"abstract":"<div><div>Flag codes, as a generalization of subspace codes, can transmit more information since the subspace channel is used many times. In this paper, we construct optimum distance flag codes of the (generalized) full admissible type <span><math><mi>t</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>,</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> on <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> with cardinality <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi><mo>−</mo><mn>1</mn></mrow></msubsup><msup><mrow><mi>q</mi></mrow><mrow><mi>i</mi><mi>k</mi><mo>+</mo><mi>h</mi></mrow></msup><mo>+</mo><mn>1</mn></math></span>, where <span><math><mi>n</mi><mo>=</mo><mi>s</mi><mi>k</mi><mo>+</mo><mi>h</mi></math></span> with <span><math><mi>s</mi><mo>≥</mo><mn>2</mn></math></span> and <span><math><mn>0</mn><mo>≤</mo><mi>h</mi><mo><</mo><mi>k</mi></math></span>. Let <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>q</mi></mrow><mrow><mi>f</mi></mrow></msubsup><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mi>D</mi></mrow><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>,</mo><mi>t</mi><mo>)</mo></math></span> denote the maximum cardinality of such codes. We provide a lower bound for this quantity. We further present a systematic construction of cardinality-consistent flag codes with larger cardinality for general flag distances. By the composition of subspace polynomials, we construct cardinality-consistent cyclic flag codes on <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> with larger cardinality than those presented in the literature.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102750"},"PeriodicalIF":1.2,"publicationDate":"2025-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416454","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}
In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.
{"title":"Normal and primitive normal elements with prescribed traces in intermediate extensions of finite fields","authors":"Arpan Chandra Mazumder , Giorgos Kapetanakis , Dhiren Kumar Basnet","doi":"10.1016/j.ffa.2025.102745","DOIUrl":"10.1016/j.ffa.2025.102745","url":null,"abstract":"<div><div>In this article, we study the existence and distribution of elements in finite field extensions with prescribed traces in several intermediate extensions that are also either normal or primitive normal. In the former case, we fully characterize the conditions under which such elements exist and provide an explicit enumeration of these elements. In the latter case we provide asymptotic results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102745"},"PeriodicalIF":1.2,"publicationDate":"2025-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416455","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 : 2025-10-23DOI: 10.1016/j.ffa.2025.102742
Farhana Kousar , Maosheng Xiong
In a recent paper [30] Zhang et al. constructed 17 families of permutation pentanomials of the form over where . In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over .
{"title":"Some permutation pentanomials over finite fields of even characteristic","authors":"Farhana Kousar , Maosheng Xiong","doi":"10.1016/j.ffa.2025.102742","DOIUrl":"10.1016/j.ffa.2025.102742","url":null,"abstract":"<div><div>In a recent paper <span><span>[30]</span></span> Zhang et al. constructed 17 families of permutation pentanomials of the form <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>(</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mi>t</mi></mrow></msup></math></span> over <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> where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In this paper for 14 of these 17 families we provide a simple explanation as to why they are permutations. We also extend these 14 families into three general classes of permutation pentanomials over <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>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102742"},"PeriodicalIF":1.2,"publicationDate":"2025-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362888","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 : 2025-10-22DOI: 10.1016/j.ffa.2025.102739
Stephen D. Cohen , Peter V. Danchev , Tomás Oliveira e Silva
We classify those finite fields whose members are the sum of an n-potent element with and a 4-potent element. It is shown that there are precisely ten non-trivial pairs for which this is the case. This continues a recent publication by Abyzov et al. (2024) [1] in which the tripotent version was examined in-depth, inasmuch as it extends recent results in this seam of research established by Abyzov and Tapkin (2024) [4].
{"title":"Finite fields whose members are the sum of a potent and a 4-potent","authors":"Stephen D. Cohen , Peter V. Danchev , Tomás Oliveira e Silva","doi":"10.1016/j.ffa.2025.102739","DOIUrl":"10.1016/j.ffa.2025.102739","url":null,"abstract":"<div><div>We classify those finite fields <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> whose members are the sum of an <em>n</em>-potent element with <span><math><mi>n</mi><mo>></mo><mn>1</mn></math></span> and a 4-potent element. It is shown that there are precisely ten non-trivial pairs <span><math><mo>(</mo><mi>q</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> for which this is the case. This continues a recent publication by Abyzov et al. (2024) <span><span>[1]</span></span> in which the tripotent version was examined in-depth, inasmuch as it extends recent results in this seam of research established by Abyzov and Tapkin (2024) <span><span>[4]</span></span>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102739"},"PeriodicalIF":1.2,"publicationDate":"2025-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362887","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}
A function from to is said to be kth order sum-free if the sum of its values over each k-dimensional -affine subspace of is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function (with defined to be 0). It is known that is 2nd order (equivalently, th order) sum-free if and only if n is odd, and it is conjectured that for , is never kth order sum-free. The conjecture has been confirmed for even n but remains open for odd n. In the present paper, we show that the conjecture holds under each of the following conditions: (1) ; (2) ; (3) ; (4) the smallest prime divisor l of n satisfies . We also determine the “right” q-ary generalization of the binary multiplicative inverse function in the context of sum-freedom. This q-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.
{"title":"On sum-free functions","authors":"Alyssa Ebeling , Xiang-dong Hou , Ashley Rydell , Shujun Zhao","doi":"10.1016/j.ffa.2025.102744","DOIUrl":"10.1016/j.ffa.2025.102744","url":null,"abstract":"<div><div>A function from <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> to <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is said to be <em>kth order sum-free</em> if the sum of its values over each <em>k</em>-dimensional <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-affine subspace of <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span> is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> (with <span><math><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> defined to be 0). It is known that <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub></math></span> is 2nd order (equivalently, <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>2</mn><mo>)</mo></math></span>th order) sum-free if and only if <em>n</em> is odd, and it is conjectured that for <span><math><mn>3</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>3</mn></math></span>, <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub></math></span> is never <em>k</em>th order sum-free. The conjecture has been confirmed for even <em>n</em> but remains open for odd <em>n</em>. In the present paper, we show that the conjecture holds under each of the following conditions: (1) <span><math><mi>n</mi><mo>=</mo><mn>13</mn></math></span>; (2) <span><math><mn>3</mn><mo>|</mo><mi>n</mi></math></span>; (3) <span><math><mn>5</mn><mo>|</mo><mi>n</mi></math></span>; (4) the smallest prime divisor <em>l</em> of <em>n</em> satisfies <span><math><mo>(</mo><mi>l</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>(</mo><mi>l</mi><mo>+</mo><mn>2</mn><mo>)</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. We also determine the “right” <em>q</em>-ary generalization of the binary multiplicative inverse function <span><math><msub><mrow><mi>f</mi></mrow><mrow><mtext>inv</mtext></mrow></msub></math></span> in the context of sum-freedom. This <em>q</em>-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102744"},"PeriodicalIF":1.2,"publicationDate":"2025-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362889","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 : 2025-10-20DOI: 10.1016/j.ffa.2025.102743
Tongliang Zhang , Lijing Zheng , Hengtai Wang , Jie Peng , Yanjun Li
Let . In a recent paper [34], Zhang and Zheng investigated several classes of permutation pentanomials of the form over with a certain linearized polynomial . They applied the multivariate method and specific techniques to analyze the number of solutions of certain equations, and proposed an open problem: the permutation property of some pentanomials of this form remains unproven. In this paper, inspired by the idea of [12], we further characterize the permutation property of such pentanomials over . The techniques presented in this paper will be useful for investigating more new classes of permutation polynomials.
{"title":"Further results on permutation pentanomials over Fq3 in characteristic two","authors":"Tongliang Zhang , Lijing Zheng , Hengtai Wang , Jie Peng , Yanjun Li","doi":"10.1016/j.ffa.2025.102743","DOIUrl":"10.1016/j.ffa.2025.102743","url":null,"abstract":"<div><div>Let <span><math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math></span>. In a recent paper <span><span>[34]</span></span>, Zhang and Zheng investigated several classes of permutation pentanomials of the form <span><math><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>0</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub></mrow></msup><mo>+</mo><mi>L</mi><mo>(</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msub><mrow><mi>ϵ</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>)</mo></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span> with a certain linearized polynomial <span><math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. They applied the multivariate method and specific techniques to analyze the number of solutions of certain equations, and proposed an open problem: the permutation property of some pentanomials of this form remains unproven. In this paper, inspired by the idea of <span><span>[12]</span></span>, we further characterize the permutation property of such pentanomials over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msub><mspace></mspace><mo>(</mo><msub><mrow><mi>d</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></math></span>. The techniques presented in this paper will be useful for investigating more new classes of permutation polynomials.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102743"},"PeriodicalIF":1.2,"publicationDate":"2025-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145362890","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 : 2025-10-17DOI: 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}