Finite Fields and Their Applications最新文献

英文中文

Points of bounded height on projective spaces over global function fields via geometry of numbers 通过数几何学研究全局函数域投影空间上的有界高点

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-04-08 DOI: 10.1016/j.ffa.2024.102417

Tristan Phillips

We give a new proof of a result of DiPippo and Wan for counting points of bounded height on projective spaces over global function fields. The new proof adapts the geometry of numbers arguments used by Schanuel in the number field case.

我们给出了 DiPippo 和 Wan 关于全局函数域上射影空间有界高点计数结果的新证明。新证明改编了数场情况下沙努埃尔使用的数的几何论证。

引用次数: 0

A correction and further improvements to the Chevalley-Warning theorems 对切瓦利-沃宁定理的修正和进一步改进

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-04-08 DOI: 10.1016/j.ffa.2024.102427

David B. Leep , Rachel L. Petrik

This paper corrects an error in the proof of Theorem 1.4 (3) of our earlier paper, Further Improvements to the Chevalley-Warning Theorems. The error originally appeared in Heath-Brown's paper, On Chevalley-Warning Theorems, which invalidates the proof of Theorem 2 (iii) in that paper. In this paper, we use a new method to give a correct proof of Theorem 1.4 (3). The correction in this paper also fixes the proof of Theorem 2 (iii) in Heath-Brown's paper. The proof in this paper provides slightly stronger estimates for some of the inequalities that were used in Further Improvements to the Chevalley-Warning Theorems.

本文纠正了我们早先的论文《对车瓦利警告定理的进一步改进》中定理 1.4 (3) 证明中的一个错误。这个错误最初出现在希斯-布朗（Heath-Brown）的论文《论车瓦利-警告定理》中，导致该论文中定理 2 (iii) 的证明无效。在本文中，我们用一种新方法给出了定理 1.4 (3) 的正确证明。本文的修正也修正了希斯-布朗论文中定理 2 (iii) 的证明。本文的证明为《对切瓦利-警告定理的进一步改进》中使用的一些不等式提供了稍强的估计。

引用次数: 0

Arithmetic crosscorrelation of binary m-sequences with coprime periods 具有共同周期的二进制 m 序列的算术相关性

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-04-06 DOI: 10.1016/j.ffa.2024.102424

Xiaoyan Jing , Keqin Feng

The arithmetic crosscorrelation of binary m-sequences with coprime periods $2^{n_{1}} - 1$ and $2^{n_{2}} - 1$ ( $\gcd (n_{1}, n_{2}) = 1$ ) is determined. The result shows that the absolute value of arithmetic crosscorrelation of such binary m-sequences is not greater than $2^{\min (n_{1}, n_{2})} - 1$ .

确定了周期为 2n1-1 和 2n2-1（gcd(n1,n2)=1）的二进制 m 序列的算术相关性。结果表明，此类二进制 m 序列算术相关性的绝对值不大于 2min(n1,n2)-1。

引用次数: 0

Bivariate functions with low c-differential uniformity 具有低 c 差均匀性的二元函数

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-04-06 DOI: 10.1016/j.ffa.2024.102422

Yanan Wu , Pantelimon Stănică , Chunlei Li , Nian Li , Xiangyong Zeng

Starting with the multiplication of elements in $F_{q}^{2}$ which is consistent with that over $F_{q^{2}}$ , where q is a prime power, via some identification of the two environments, we investigate the c-differential uniformity for bivariate functions $F (x, y) = (G (x, y), H (x, y))$ . By carefully choosing the functions $G (x, y)$ and $H (x, y)$ , we present several constructions of bivariate functions with low c-differential uniformity, in particular, many PcN and APcN functions can be produced from our constructions.

从 Fq2 中元素的乘法与 Fq2 上元素的乘法一致（其中 q 是质幂）开始，通过对两种环境的一些识别，我们研究了二元函数 F(x,y)=(G(x,y),H(x,y)) 的 c 微分均匀性。通过精心选择函数 G(x,y) 和 H(x,y)，我们提出了几种具有低 c 差均匀性的二元函数构造，特别是，许多 PcN 和 APcN 函数可以从我们的构造中产生。

引用次数: 0

Hamming distances of constacyclic codes of length 7ps over Fpm Fpm 上长度为 7ps 的常环码的汉明距离

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-04-05 DOI: 10.1016/j.ffa.2024.102420

Hai Q. Dinh , Hieu V. Ha , Nhan T.V. Nguyen , Nghia T.H. Tran , Thieu N. Vo

In this paper, we study constacyclic codes of length $n = 7 p^{s}$ over a finite field of characteristics p, where $p \neq 7$ is an odd prime number and s a positive integer. The previous methods in the literature that were used to compute the Hamming distances of repeated-root constacyclic codes of lengths $n p^{s}$ with $1 \leq n \leq 6$ cannot be applied to completely determine the Hamming distances of those with $n = 7$ . This is due to the high computational complexity involved and the large number of unexpected intermediate results that arise during the computation. To overcome this challenge, we propose a computer-assisted method for determining the Hamming distances of simple-root constacyclic codes of length 7, and then utilize it to derive the Hamming distances of the repeated-root constacyclic codes of length $7 p^{s}$ . Our method is not only straightforward to implement but also efficient, making it applicable to these codes with larger values of n as well. In addition, all self-orthogonal, dual-containing, self-dual, MDS and AMDS codes among them will also be characterized.

本文研究在有限特征域 p 上长度为 n=7ps 的常环码，其中 p≠7 是奇素数，s 是正整数。以往文献中用于计算长度为 nps 的重复根常簇码的汉明距离（1≤n≤6）的方法无法完全确定长度为 n=7 的常簇码的汉明距离。这是因为涉及的计算复杂度很高，而且在计算过程中会出现大量意想不到的中间结果。为了克服这一难题，我们提出了一种计算机辅助方法，用于确定长度为 7 的单根共环码的汉明距离，然后利用它推导出长度为 7ps 的重复根共环码的汉明距离。我们的方法不仅简单易行，而且效率很高，因此也适用于这些 n 值较大的编码。此外，我们还将对其中的所有自正交码、双含码、自双码、MDS 码和 AMDS 码进行表征。

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引用次数: 0

On trivial cyclically covering subspaces of Fqn 关于 Fqn 的琐碎循环覆盖子空间

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-03-29 DOI: 10.1016/j.ffa.2024.102423

Jing Huang

A subspace of $F_{q}^{n}$ is called a cyclically covering subspace if for every vector of $F_{q}^{n}$ , operating a certain number of cyclic shifts on it, the resulting vector lies in the subspace. In this paper, we study the problem of under what conditions $F_{q}^{n}$ is itself the only covering subspace of $F_{q}^{n}$ , symbolically, $h_{q} (n) = 0$ , which is an open problem posed in Cameron et al. (2019) [3] and Aaronson et al. (2021) [1]. We apply the primitive idempotents of the cyclic group algebra to attack this problem; when q is relatively prime to n, we obtain a necessary and sufficient condition under which $h_{q} (n) = 0$ , which completely answers the problem in this case. Our main result reveals that the problem can be fully reduced to that of determining the values of the trace function over finite fields. As consequences, we explicitly determine several infinitely families of $F_{q}^{n}$ which satisfy $h_{q} (n) = 0$ .

如果对 Fqn 的每个向量进行一定次数的循环移位，得到的向量位于该子空间中，则 Fqn 的一个子空间称为循环覆盖子空间。本文研究的问题是，在什么条件下 Fqn 本身是 Fqn 的唯一覆盖子空间，符号为 hq(n)=0，这是 Cameron 等（2019）[3] 和 Aaronson 等（2021）[1] 提出的一个开放问题。我们应用循环群代数的基元幂级数来解决这个问题；当 q 相对于 n 是素数时，我们得到了 hq(n)=0 的必要条件和充分条件，在这种情况下完全解答了这个问题。我们的主要结果表明，这个问题完全可以简化为确定有限域上的迹函数值。因此，我们明确地确定了满足 hq(n)=0 的 Fqn 的几个无穷族。

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引用次数: 0

Constructing permutation polynomials from permutation polynomials of subfields 从子域的置换多项式构建置换多项式

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-03-29 DOI: 10.1016/j.ffa.2024.102415

Lucas Reis , Qiang Wang

<div>In this paper we study the permutational property of polynomials of the form <math><mi>f</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>k</mi><mo>(</mo><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>⋅</mo><mi>M</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math> over the finite field <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math>, where <math><mi>L</mi><mo>,</mo><mi>M</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math> are q-linearized polynomials and <math><mi>k</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math> satisfies a generic condition. We specialize in the case where <math><mi>L</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is the linearized q-associate of <math><msub><mrow><mi>g</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>a</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>−</mo><mi>a</mi><mo>)</mo></math>, t is a divisor of n and <math><mi>a</mi><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> satisfies <math><msup><mrow><mi>a</mi></mrow><mrow><mi>n</mi><mo>/</mo><mi>t</mi></mrow></msup><mo>=</mo><mn>1</mn></math>. This unifies many recent explicit constructions and provides new explicit constructions of permutation polynomials and their inverses. Moreover, we introduce a new algorithmic method to produce many permutation polynomials of <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math> from permutations of <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub></math>, by simply solving a system of independent equations of the form <math><msub><mrow><mi>Tr</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><msup><mrow><mi>q</mi></mrow><mrow><mi>t</mi></mrow></msup></mrow></msub><mo>(</mo><msup><mrow><mi>δ</mi></mrow><mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msup><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mi>i</mi></mrow></msub></math>, where the <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub></math>'s are the coefficients of f. In fact, the same method can be

本文研究有限域 Fqn 上 f(L(x))+k(L(x))⋅M(x)∈Fqn[x]形式的多项式的置换性质，其中 L,M∈Fq[x] 是 q 线性多项式，k∈Fqn[x] 满足一般条件。我们专门研究 L(x) 是 gt,a(x)=(xn-1)/(xt-a) 的线性化 q 关联，t 是 n 的除数，且 a∈Fq 满足 an/t=1 的情况。这统一了许多最新的显式构造，并提供了关于置换多项式及其倒数的新显式构造。此外，我们还介绍了一种新的算法方法，通过简单地求解形式为 Trqn/qt(δi-1ai)=ci（其中 ai 是 f 的系数）的独立方程组，就能从 Fqt 的置换中生成许多 Fqn 的置换多项式。

引用次数: 0

Application of the Cartier operator in coding theory 卡蒂埃算子在编码理论中的应用

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-03-29 DOI: 10.1016/j.ffa.2024.102419

Vahid Nourozi

The a-number is an invariant of the isomorphism class of the p-torsion group scheme. We use the Cartier operator on $H^{0} (A_{2}, Ω^{1})$ to find a closed formula for the a-number of the form $A_{2} = v (Y^{\sqrt{q}} + Y - x^{\frac{\sqrt{q} + 1}{2}})$ where $q = p^{s}$ over the finite field $F_{q^{2}}$ . The application of the computed a-number in coding theory is illustrated by the relationship between the algebraic properties of the curve and the parameters of codes that are supported by it.

a 数是 p 扭转群方案同构类的不变式。我们利用 H0(A2,Ω1)上的卡蒂埃算子，找到了有限域 Fq2 上 q=ps 的 A2=v(Yq+Y-xq+12) 形式的 a 数封闭公式。曲线的代数特性与曲线支持的编码参数之间的关系，说明了计算出的 a 数在编码理论中的应用。

引用次数: 0

Private information retrieval from locally repairable databases with colluding servers 从有串通服务器的本地可修复数据库中检索私人信息

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-03-28 DOI: 10.1016/j.ffa.2024.102421

Umberto Martínez-Peñas

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g, locality r, local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of $r + δ - 1$ ) are downloaded per server. The PIR scheme achieves the (download) rate $R = (N - k - r t + 1) / N$ , where $N = g r$ is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that $k + r t \leq N$ . For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications.

我们考虑从有串通服务器的编码数据库中进行信息论私人信息检索（PIR）。我们首次将本地可修复存储代码（LRC）作为研究对象。我们的主要贡献是基于线性化里德-所罗门（Reed-Solomon）编码的最大可恢复（MR）LRC 的 PIR 方案，该方案在许多参数机制下实现了 MR-LRC 中最小的字段大小。在我们的方案中，节点由编码词符号标识，服务器由节点的本地组标识。每个服务器只下载本地非冗余信息，即每个服务器只下载 r 个节点（r+δ-1 中的 r 个）。PIR 方案的（下载）速率为 R=（N-k-rt+1）/N，其中 N=gr 为去除局部奇偶校验后得到的 MDS 代码长度，且对于任意 t 个串通服务器，k+rt≤N。对于不受限制的存储文件数量，所获得的速率严格大于那些适用于任何 MDS 代码的已知 PIR 方案。最后，当用户和每个服务器之间的通信是通过线性网络编码完成时，所获得的 PIR 方案也可以进行调整，在这种情况下获得与以前的 PIR 方案相同的速率，但有限场大小是多项式，而不是指数。我们的速率与里德-所罗门编码的 PIR 方案相当，但里德-所罗门编码与 MR-LRC 特性或线性网络编码不兼容，因此我们的 PIR 方案在应用上限制较少。

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引用次数: 0

Primitive elements of finite fields Fqr avoiding affine hyperplanes for q = 4 and q = 5 q = 4 和 q = 5 时避免仿射超平面的有限域 Fqr 的基元

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-03-26 DOI: 10.1016/j.ffa.2024.102416

Philipp A. Grzywaczyk , Arne Winterhof

For a finite field $F_{q^{r}}$ with fixed q and r sufficiently large, we prove the existence of a primitive element outside of a set of r many affine hyperplanes for $q = 4$ and $q = 5$ . This complements earlier results by Fernandes and Reis for $q \geq 7$ . For $q = 3$ the analogous result can be derived from a very recent bound on character sums of Iyer and Shparlinski. For $q = 2$ the set consists only of a single element, and such a result is thus not possible.

对于固定 q 和 r 足够大的有限域 Fqr，我们证明了在 q=4 和 q=5 时，在一组 r 个仿射超平面之外存在一个基元。这是对费尔南德斯和雷斯早先关于 q≥7 的结果的补充。对于 q=3 ，类似的结果可以从 Iyer 和 Shparlinski 最新的特征和约束中推导出来。对于 q=2 ，集合只包含一个元素，因此不可能得出这样的结果。

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