Finite Fields and Their Applications最新文献

{"title":"On a class of complete permutation quadrinomials","authors":"Chin Hei Chan ,&nbsp;Zhiguo Ding ,&nbsp;Nian Li ,&nbsp;Xi Xie ,&nbsp;Maosheng Xiong ,&nbsp;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":"2026-02-01","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}

{"title":"Estimates on the number of rational solutions of Markoff-Hurwitz equations over finite fields","authors":"Miriam Abdón ,&nbsp;Daniela Alves de Oliveira ,&nbsp;Juliane Capaverde ,&nbsp;Mariana Pérez ,&nbsp;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":"2026-02-01","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}

{"title":"Construction of three class of at most four-weight binary linear codes and their applications","authors":"Tonghui Zhang ,&nbsp;Pinhui Ke ,&nbsp;Zuling Chang","doi":"10.1016/j.ffa.2025.102754","DOIUrl":"10.1016/j.ffa.2025.102754","url":null,"abstract":"<div><div>Three classes of binary linear codes with at most four nonzero weights were constructed in this paper, in which two of them are projective three-weight codes. As applications, <em>s</em>-sum sets for any odd <span><math><mi>s</mi><mo>&gt;</mo><mn>1</mn></math></span> were constructed.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102754"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466207","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Large cyclic subspace codes over finite fields","authors":"He Zhang ,&nbsp;Chunming Tang ,&nbsp;Xiwang Cao ,&nbsp;Gaojun Luo","doi":"10.1016/j.ffa.2025.102722","DOIUrl":"10.1016/j.ffa.2025.102722","url":null,"abstract":"<div><div>Cyclic subspace codes play a crucial role in random network coding. Designing such cyclic subspace codes with the largest possible code size and minimum distance remains a classical problem. Roth et al. (2018) <span><span>[28]</span></span> first investigated optimal cyclic subspace codes via Sidon spaces and proved that the orbit of a Sidon space is an optimal cyclic subspace code with full-length orbit. This paper introduces a new method, namely the intermediate extension field, to construct Sidon spaces and cyclic subspace codes. The main results show that our new codes over intermediate fields have optimal minimum distance and contain more codewords than known constructions. Therefore, this work improves the lower bound of optimal cyclic subspace codes.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102722"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145011304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the study of semi-involutory and semi-orthogonal matrices","authors":"Yogesh Kumar ,&nbsp;Susanta Samanta ,&nbsp;P.R. Mishra ,&nbsp;Atul Gaur","doi":"10.1016/j.ffa.2025.102730","DOIUrl":"10.1016/j.ffa.2025.102730","url":null,"abstract":"<div><div>This paper investigates semi-involutory and semi-orthogonal matrices, presenting two algorithms for verifying these properties for <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math></span>. The algorithms significantly reduce computational complexity by avoiding the need for non-singular diagonal matrices. The structure of circulant matrices with these properties is also explored, including the derivation of the exact form of their corresponding diagonal matrices. We further investigate MDS matrices with semi-involutory and semi-orthogonal properties. We prove that semi-involutory circulant matrices of order <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>=</mo><mn>1</mn></math></span>, cannot be MDS matrices. Additionally, we show that semi-orthogonal circulant matrices of order <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span> cannot be MDS. Finally, explicit formulas for counting <span><math><mn>3</mn><mo>×</mo><mn>3</mn></math></span> semi-involutory and semi-orthogonal MDS matrices over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span> are derived, and exact counts for <span><math><mn>4</mn><mo>×</mo><mn>4</mn></math></span> semi-involutory and semi-orthogonal MDS matrices over <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></mrow></msub></math></span> for <span><math><mi>m</mi><mo>=</mo><mn>3</mn></math></span> and 4 are provided.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102730"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145221460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Some permutation pentanomials over finite fields of even characteristic","authors":"Farhana Kousar ,&nbsp;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":"2026-02-01","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}

{"title":"The multiplicity-one theorem for the superspeciality of curves of genus two","authors":"Shushi Harashita,&nbsp;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":"2026-02-01","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}

{"title":"Linear l-intersection pairs of matrix-product codes and their applications","authors":"Xiujing Zheng,&nbsp;Sujuan Huang,&nbsp;Shixin Zhu","doi":"10.1016/j.ffa.2025.102758","DOIUrl":"10.1016/j.ffa.2025.102758","url":null,"abstract":"<div><div>The linear <em>l</em>-intersection pairs of codes are a generalization of linear complementary dual (LCD) codes, hulls of codes and linear complementary pairs (LCPs) of codes. Matrix-product codes are extended versions derived from shorter codes through matrix-product techniques. In this paper, we investigate linear <em>l</em>-intersection pairs of matrix-product codes. The characterization of these pairs can be achieved by examining the dimension of the intersection between their respective constituent codes. For the dimension part of the conjecture for linear <em>l</em>-intersection pairs of codes proposed by Guenda et al. (Des Codes Cryptogr. <strong>88</strong>: 133-152, 2020), we prove that if the conjecture holds for prime lengths, then its dimension part holds. As a practical application, linear <em>l</em>-intersection pairs of matrix-product codes are utilized to the constructions of asymmetric quantum error-correcting (AQEC) codes and asymmetric entanglement-assisted quantum error-correcting (AEAQEC) codes. Some instances exhibit favorable parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102758"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145519892","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 S-complete mappings for large S","authors":"Robert S. Coulter ,&nbsp;Paul Hearding","doi":"10.1016/j.ffa.2025.102756","DOIUrl":"10.1016/j.ffa.2025.102756","url":null,"abstract":"<div><div>Given a subset <em>S</em> of a finite field, an <em>S</em>-complete mapping is a polynomial <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> for which <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>+</mo><mi>c</mi><mi>X</mi></math></span> is a permutation polynomial over the finite field for each <span><math><mi>c</mi><mo>∈</mo><mi>S</mi></math></span>. In this paper, we introduce a new method for constructing permutation polynomials and use it to establish a class of <em>S</em>-complete mappings with “large” <em>S</em>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102756"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466211","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":"Permutation polynomials and finite projective spaces","authors":"Tong Lin ,&nbsp;Qiang Wang","doi":"10.1016/j.ffa.2025.102752","DOIUrl":"10.1016/j.ffa.2025.102752","url":null,"abstract":"<div><div>Using arbitrary bases for the finite field <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> over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span>, we obtain the generalized Möbius transformations (GMTs), which are a class of bijections between the projective geometry <span><math><mtext>PG</mtext><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span> and the set of roots of unity <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msub><mo>⊆</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math></span>, where <span><math><mi>n</mi><mo>≥</mo><mn>2</mn></math></span> is any integer. We also introduce a class of projective polynomials, using the properties of which we determine the inverses of the GMTs. Moreover, we study the roots of those projective polynomials, which lead to a three-way correspondence between partitions of <span><math><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup><mo>,</mo><msub><mrow><mi>μ</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msub></math></span> and <span><math><mtext>PG</mtext><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>q</mi><mo>)</mo></math></span>. Through this correspondence and the GMTs, we construct permutation polynomials of index <span><math><mfrac><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow><mrow><mi>q</mi><mo>−</mo><mn>1</mn></mrow></mfrac></math></span> over <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>.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"110 ","pages":"Article 102752"},"PeriodicalIF":1.2,"publicationDate":"2026-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145466210","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}