Pub Date : 2025-09-08DOI: 10.1016/j.ffa.2025.102722
He Zhang , Chunming Tang , Xiwang Cao , Gaojun Luo
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) [28] 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.
循环子空间码在随机网络编码中起着至关重要的作用。设计尽可能大码长和最小距离的循环子空间码仍然是一个经典问题。Roth et al.(2018)[28]首次通过西顿空间研究了最优循环子空间码，证明了西顿空间的轨道是具有全长轨道的最优循环子空间码。本文介绍了构造西顿空间和循环子空间码的一种新方法，即中间可拓域。主要结果表明，我们的新码在中间域上具有最佳的最小距离，并且比已知结构包含更多的码字。因此，本文改进了最优循环子空间码的下界。
{"title":"Large cyclic subspace codes over finite fields","authors":"He Zhang , Chunming Tang , Xiwang Cao , 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":"2025-09-08","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}
Pub 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":"2025-09-04","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}
MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85%, which is much higher than previously reported results.
{"title":"Construction of MDS Euclidean self-dual codes via multiple subsets","authors":"Weirong Meng , Weijun Fang , Fang-Wei Fu , Haiyan Zhou , Ziyi Gu","doi":"10.1016/j.ffa.2025.102718","DOIUrl":"10.1016/j.ffa.2025.102718","url":null,"abstract":"<div><div>MDS self-dual codes have good algebraic structure, and their parameters are completely determined by the code length. In recent years, the construction of MDS Euclidean self-dual codes with new lengths has become an important issue in coding theory. In this paper, we are committed to constructing new MDS Euclidean self-dual codes via generalized Reed-Solomon (GRS) codes and their extended (EGRS) codes. The main effort of our constructions is to find suitable subsets of finite fields as the evaluation sets, ensuring that the corresponding (extended) GRS codes are Euclidean self-dual. Firstly, we present a method for selecting evaluation sets from multiple intersecting subsets and provide a theorem to guarantee that the chosen evaluation sets meet the desired criteria. Secondly, based on this theorem, we construct six new classes of MDS Euclidean self-dual codes using the norm function, as well as the union of three multiplicity subgroups and their cosets respectively. Finally, in our constructions, the proportion of possible MDS Euclidean self-dual codes exceeds 85%, which is much higher than previously reported results.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102718"},"PeriodicalIF":1.2,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144912895","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 : 2025-08-21DOI: 10.1016/j.ffa.2025.102715
Derek Garton , Jeffrey Lin Thunder , Colin Weir
In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus g defined over a finite field with a given a-number. In characteristic three this method gives exact probabilities for curves of the form with monic and cubefree—probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of q) for these probabilities when restricting to squarefree f. As a consequence, for positive integers a and g we show that the nonempty strata of the moduli space of hyperelliptic curves of genus g consisting of those curves with a-number a are of codimension . This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are . Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.—one that matches the available data.
{"title":"The distribution of a-numbers of hyperelliptic curves in characteristic three","authors":"Derek Garton , Jeffrey Lin Thunder , Colin Weir","doi":"10.1016/j.ffa.2025.102715","DOIUrl":"10.1016/j.ffa.2025.102715","url":null,"abstract":"<div><div>In this paper we present a new approach to counting the proportion of hyperelliptic curves of genus <em>g</em> defined over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></span> with a given <em>a</em>-number. In characteristic three this method gives exact probabilities for curves of the form <span><math><msup><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo></math></span> with <span><math><mi>f</mi><mo>(</mo><mi>X</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math></span> monic and cubefree—probabilities that match the data presented by Cais et al. in previous work. These results are sufficient to derive precise estimates (in terms of <em>q</em>) for these probabilities when restricting to squarefree <em>f</em>. As a consequence, for positive integers <em>a</em> and <em>g</em> we show that the nonempty strata of the moduli space of hyperelliptic curves of genus <em>g</em> consisting of those curves with <em>a</em>-number <em>a</em> are of codimension <span><math><mn>2</mn><mi>a</mi><mo>−</mo><mn>1</mn></math></span>. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are <span><math><mi>a</mi><mo>(</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math></span>. Finally, our results allow for an alternative heuristic conjecture to that of Cais et al.—one that matches the available data.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102715"},"PeriodicalIF":1.2,"publicationDate":"2025-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144879454","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 : 2025-08-13DOI: 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":"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}
Pub 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":"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}
Pub Date : 2025-08-12DOI: 10.1016/j.ffa.2025.102716
Gerardo Vega
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.
{"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}
Pub Date : 2025-08-04DOI: 10.1016/j.ffa.2025.102712
Ramy Taki Eldin , Patrick Solé
In this study, we consider the Euclidean and Galois hulls of multi-twisted (MT) codes over a finite field of characteristic p. Let G be a generator polynomial matrix (GPM) of an MT code . For any , the κ-Galois hull of , denoted by , is the intersection of with its κ-Galois dual. The main result in this paper is that a GPM for has been obtained from G. We start by associating a linear code with G. We show that is quasi-cyclic. In addition, we prove that the dimension of is the difference between the dimension of and that of . Thus the determinantal divisors are used to derive a formula for the dimension of . Finally, we deduce a GPM formula for . In particular, we handle the cases of κ-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.
{"title":"Generator polynomial matrices of the Galois hulls of multi-twisted codes","authors":"Ramy Taki Eldin , 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><</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}
Pub Date : 2025-07-31DOI: 10.1016/j.ffa.2025.102710
Xi Xie , Nian Li , Qiang Wang , Xiangyong Zeng , Yinglong Du
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 -functions. First, by refining specific exponential sums, we propose two classes of power functions over with low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of -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 -functions with low DLU, including some that achieve optimal or near-optimal DLU compared to existing results.
{"title":"Construction of (n,n)-functions with low differential-linear uniformity","authors":"Xi Xie , Nian Li , Qiang Wang , Xiangyong Zeng , 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}
Pub Date : 2025-07-29DOI: 10.1016/j.ffa.2025.102711
Na Chen, Haiyan Zhou
Let be a Galois ring of characteristic with cardinality , where p is a prime. Let , , and denote the number of solutions of the equations , , and , respectively. In this paper, we show that for any , the generating functions
{"title":"The number of solutions of diagonal cubic equations over Galois rings GR(p2,r)","authors":"Na Chen, 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}
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