Finite Fields and Their Applications最新文献_第9页

Corrigendum to “Higher Grassmann codes II” [Finite Fields Appl. 89 (2023) 102211] “高等格拉斯曼规范II”的勘误表[有限域应用89 (2023)102211]

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-07-17 DOI: 10.1016/j.ffa.2025.102700

Mahir Bilen Can , Roy Joshua , G.V. Ravindra

引用次数: 0

New constructions of permutation polynomials of the form x+γTrqq3(h(x)) over finite fields with even characteristic 偶特征有限域上形式为x+γTrqq3（h(x)）的置换多项式的新构造

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-07-09 DOI: 10.1016/j.ffa.2025.102694

Sha Jiang, Kangquan Li, Longjiang Qu

引用次数: 0

Generator polynomial matrices of the Galois hulls of multi-twisted codes 多扭曲码伽罗瓦壳的生成多项式矩阵

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-08-04 DOI: 10.1016/j.ffa.2025.102712

Ramy Taki Eldin , Patrick Solé

{"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 <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>e</mi></mrow></msup></mrow></msub></math> of characteristic p. Let G be a generator polynomial matrix (GPM) of an MT code <math><mi>C</mi></math>. For any <math><mn>0</mn><mo>≤</mo><mi>κ</mi><mo><</mo><mi>e</mi></math>, the κ-Galois hull of <math><mi>C</mi></math>, denoted by <math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math>, is the intersection of <math><mi>C</mi></math> with its κ-Galois dual. The main result in this paper is that a GPM for <math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math> has been obtained from G. We start by associating a linear code <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>G</mi></mrow></msub></math> with G. We show that <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>G</mi></mrow></msub></math> is quasi-cyclic. In addition, we prove that the dimension of <math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math> is the difference between the dimension of <math><mi>C</mi></math> and that of <math><msub><mrow><mi>Q</mi></mrow><mrow><mi>G</mi></mrow></msub></math>. Thus the determinantal divisors are used to derive a formula for the dimension of <math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math>. Finally, we deduce a GPM formula for <math><msub><mrow><mi>h</mi></mrow><mrow><mi>κ</mi></mrow></msub><mrow><mo>(</mo><mi>C</mi><mo>)</mo></mrow></math>. 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.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102712"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","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}

引用次数: 0

Construction of (n,n)-functions with low differential-linear uniformity 低微分线性均匀性（n,n）函数的构造

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-07-31 DOI: 10.1016/j.ffa.2025.102710

Xi Xie , Nian Li , Qiang Wang , Xiangyong Zeng , Yinglong Du

{"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 <math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math>-functions. First, by refining specific exponential sums, we propose two classes of power functions over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math> with low differential-linear uniformity (DLU). Next, we further investigate the differential-linear properties of <math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math>-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 <math><mo>(</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>)</mo></math>-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":"2026-01-01","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}

引用次数: 0

Construction of MDS Euclidean self-dual codes via multiple subsets 基于多子集的MDS欧几里德自对偶码的构造

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-08-29 DOI: 10.1016/j.ffa.2025.102718

Weirong Meng , Weijun Fang , Fang-Wei Fu , Haiyan Zhou , Ziyi Gu

{"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":"2026-01-01","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}

引用次数: 0

Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d （r,δ）-定值q和d的局部可修码的大小界限

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-07-11 DOI: 10.1016/j.ffa.2025.102697

Wenqin Zhang , Chen Yuan , Yuan Luo , Nian Li

{"title":"Bounds on the size of (r,δ)-locally repairable codes for fixed values q and d","authors":"Wenqin Zhang , Chen Yuan , Yuan Luo , Nian Li","doi":"10.1016/j.ffa.2025.102697","DOIUrl":"10.1016/j.ffa.2025.102697","url":null,"abstract":"<div><div>Erasure codes can strengthen fault-tolerance and reliability in distributed storage systems. One of these, locally repairable codes (LRCs), plays a crucial role. A locally repairable code with locality r (called r-LRC) can recover any coded symbol by accessing at most r other coded symbols. The original definition of LRCs is used to repair a single failed node. To address the practical concern of multiple node failures, the concept of LRCs with locality <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math> was introduced by Prakash et al. (2012) which can be seen as a generalization of r-LRCs. Since then, the bounds and constructions of <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs have been extensively studied.</div><div>This paper is dedicated to both the bounds and constructions of <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs with disjoint local repair groups over a finite field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>, where the parameters r, δ, the alphabet size q, and the minimum distance d are fixed constants, while the code length n tends to infinity. Inspired by the method of the classical Gilbert-Varshamov (GV) bound, we first derive an asymptotic Gilbert-Varshamov-type bound for <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs in this regime. We manage to show that this GV-type bound works as a threshold for random linear <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs with disjoint local repair groups using the first and second moment methods. As a corollary, such a random linear <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRC has a high probability of attaining the GV-type bound. As an analogue to the classic GV-type bound, we present two constructions of <math><mo>(</mo><mi>r</mi><mo>,</mo><mi>δ</mi><mo>)</mo></math>-LRCs that each beats this GV-type bound. One construction is obtained from a straightforward concatenation of an outer BCH code and an inner <math><mo>[</mo><mi>r</mi><mo>+</mo><mi>δ</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>r</mi><mo>]</mo></math> MDS code. Another construction is based on the Kronecker product of two matrices. To complement our results, the case that n is large but finite is also considered. In this regime, we provide an explicit upper bound for the binary r-LRCs, which is an improvement over the one in Ma and Ge (2019). Furthermore, this bound is shown to be tight for some specific parameters.</div></div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":"109 ","pages":"Article 102697"},"PeriodicalIF":1.2,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144596550","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}

引用次数: 0

Nullities of multisets and value sets of multivariate polynomials 多元多项式的多集和值集的零值

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-07-23 DOI: 10.1016/j.ffa.2025.102709

Wei Cao

引用次数: 0

The number of solutions of diagonal cubic equations over Galois rings GR(p2,r) 伽罗瓦环上对角三次方程的解数

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-07-29 DOI: 10.1016/j.ffa.2025.102711

Na Chen, Haiyan Zhou

引用次数: 0

A characterization and an explicit description of all primitive polynomials of degree two 二阶多项式的一个特征和一个明确的描述

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-08-12 DOI: 10.1016/j.ffa.2025.102716

Gerardo Vega

引用次数: 0

The distribution of a-numbers of hyperelliptic curves in characteristic three 特征3超椭圆曲线的a数分布

IF 1.2 3区数学

Finite Fields and Their Applications Pub Date : 2026-01-01 Epub Date: 2025-08-21 DOI: 10.1016/j.ffa.2025.102715

Derek Garton , Jeffrey Lin Thunder , Colin Weir

{"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 g defined over a finite field <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> with a given a-number. In characteristic three this method gives exact probabilities for curves of the form <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> with <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> 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 <math><mn>2</mn><mi>a</mi><mo>−</mo><mn>1</mn></math>. This contrasts with the analogous result for the moduli space of abelian varieties in which the codimensions of the strata are <math><mi>a</mi><mo>(</mo><mi>a</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></math>. 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":"2026-01-01","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}

引用次数: 0