Finite Fields and Their Applications最新文献

{"title":"Irreducibility of polynomials with square coefficients over finite fields","authors":"Lior Bary-Soroker,&nbsp;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":"2025-08-13","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}

{"title":"Local permutation polynomials and their companions","authors":"Sartaj Ul Hasan,&nbsp;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":"2025-08-12","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}

{"title":"A characterization and an explicit description of all primitive polynomials of degree two","authors":"Gerardo Vega","doi":"10.1016/j.ffa.2025.102716","DOIUrl":"10.1016/j.ffa.2025.102716","url":null,"abstract":"<div><div>For polynomials of degree two over finite fields, we present an improvement of Fitzgerald's characterization of primitive polynomials. We then use this new characterization to obtain an explicit, complete, and simple description of all primitive polynomials of degree two over finite fields.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102716"},"PeriodicalIF":1.2,"publicationDate":"2025-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144827468","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":"Generator polynomial matrices of the Galois hulls of multi-twisted codes","authors":"Ramy Taki Eldin ,&nbsp;Patrick Solé","doi":"10.1016/j.ffa.2025.102712","DOIUrl":"10.1016/j.ffa.2025.102712","url":null,"abstract":"<div><div>In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></mrow></msub></math></span> of characteristic <em>p</em>. Let <strong>G</strong> be a generator polynomial matrix (GPM) of an MT code <span><math><mi>C</mi></math></span>. For any <span><math><mn>0</mn><mo>≤</mo><mi>κ</mi><mo>&lt;</mo><mi>e</mi></math></span>, the <em>κ</em>-Galois hull of <span><math><mi>C</mi></math></span>, denoted by <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>, is the intersection of <span><math><mi>C</mi></math></span> with its <em>κ</em>-Galois dual. The main result in this paper is that a GPM for <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span> has been obtained from <strong>G</strong>. We start by associating a linear code <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> with <strong>G</strong>. We show that <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> is quasi-cyclic. In addition, we prove that the dimension of <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span> is the difference between the dimension of <span><math><mi>C</mi></math></span> and that of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>. Thus the determinantal divisors are used to derive a formula for the dimension of <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>. Finally, we deduce a GPM formula for <span><math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math></span>. In particular, we handle the cases of <em>κ</em>-Galois self-orthogonal and linear complementary dual MT codes; we establish equivalent conditions that characterize these cases. Equivalent results can be deduced immediately for the classes of cyclic, constacyclic, quasi-cyclic, generalized quasi-cyclic, and quasi-twisted codes, because they are all special cases of MT codes. Some numerical examples, containing codes with the best-known parameters, are used to illustrate the theoretical results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102712"},"PeriodicalIF":1.2,"publicationDate":"2025-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144772334","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 (n,n)-functions with low differential-linear uniformity","authors":"Xi Xie ,&nbsp;Nian Li ,&nbsp;Qiang Wang ,&nbsp;Xiangyong Zeng ,&nbsp;Yinglong Du","doi":"10.1016/j.ffa.2025.102710","DOIUrl":"10.1016/j.ffa.2025.102710","url":null,"abstract":"<div><div>The differential-linear connectivity table (DLCT), introduced by Bar-On et al. at EUROCRYPT'19, is a novel tool that captures the dependency between the two subciphers involved in differential-linear attacks. This paper is devoted to exploring the differential-linear properties of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-functions. First, by refining specific exponential sums, we propose two classes of power functions over <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> with low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-functions that are polynomials by utilizing power functions with known DLU. Specifically, by combining a cubic function with quadratic functions, and employing generalized cyclotomic mappings, we construct several classes of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>-functions with low DLU, including some that achieve optimal or near-optimal DLU compared to existing results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102710"},"PeriodicalIF":1.2,"publicationDate":"2025-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144749393","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 number of solutions of diagonal cubic equations over Galois rings GR(p2,r)","authors":"Na Chen,&nbsp;Haiyan Zhou","doi":"10.1016/j.ffa.2025.102711","DOIUrl":"10.1016/j.ffa.2025.102711","url":null,"abstract":"<div><div>Let <span><math><mi>R</mi><mo>=</mo><mi>G</mi><mi>R</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mi>r</mi><mo>)</mo></math></span> be a Galois ring of characteristic <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> with cardinality <span><math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mi>r</mi></mrow></msup></math></span>, where <em>p</em> is a prime. Let <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span>, <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>z</mi><mo>)</mo></math></span>, <span><math><msubsup><mrow><mi>A</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>z</mi><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>z</mi><mo>)</mo></math></span> denote the number of solutions of the equations <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mi>z</mi></math></span>, <span><math><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>z</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math></span>, <span><math><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mi>z</mi></math></span> and <span><math><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mo>…</mo><mo>+</mo><mi>p</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>3</mn></mrow></msubsup><mo>+</mo><mi>z</mi><msubsup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>3</mn></mrow></msubsup><mo>=</mo><mn>0</mn></math></span>, respectively. In this paper, we show that for any <span><math><mi>z</mi><mo>∈</mo><mi>R</mi></math></span>, the generating functions <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msub","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102711"},"PeriodicalIF":1.2,"publicationDate":"2025-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144722080","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":"Nullities of multisets and value sets of multivariate polynomials","authors":"Wei Cao","doi":"10.1016/j.ffa.2025.102709","DOIUrl":"10.1016/j.ffa.2025.102709","url":null,"abstract":"<div><div>We extend the nullity for a finite 1-tuple multiset, which was introduced by Nica, to a finite <em>m</em>-tuple multiset, and then use it to give an upper bound for the value set of a multivariate polynomial over the multisets drawn from a field. Our results generalize and refine two generalizations of original Wan's upper bound for the value set of a univariate polynomial in finite fields.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102709"},"PeriodicalIF":1.2,"publicationDate":"2025-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144686900","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":"Weierstrass semigroups and automorphism group of a maximal function field with the third largest possible genus, q ≡ 1 (mod 3)","authors":"Peter Beelen ,&nbsp;Maria Montanucci ,&nbsp;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":"2025-07-21","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}

{"title":"Corrigendum to “Higher Grassmann codes II” [Finite Fields Appl. 89 (2023) 102211]","authors":"Mahir Bilen Can ,&nbsp;Roy Joshua ,&nbsp;G.V. Ravindra","doi":"10.1016/j.ffa.2025.102700","DOIUrl":"10.1016/j.ffa.2025.102700","url":null,"abstract":"","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102700"},"PeriodicalIF":1.2,"publicationDate":"2025-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144653484","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 iso-dual MDS codes from elliptic curves","authors":"Yunlong Zhu ,&nbsp;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":"2025-07-16","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}