Finite Fields and Their Applications最新文献

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Two classes of LCD BCH codes over finite fields 有限域上的两类 LCD BCH 码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-31 DOI: 10.1016/j.ffa.2024.102478

Yuqing Fu , Hongwei Liu

BCH codes form a special subclass of cyclic codes and have been extensively studied in the past decades. Determining the parameters of BCH codes, however, has been an important but difficult problem. Recently, in order to further investigate the dual codes of BCH codes, the concept of dually-BCH codes was proposed. In this paper, we study BCH codes of lengths $\frac{q^{m} + 1}{q + 1}$ and $q^{m} + 1$ over the finite field $F_{q}$ , both of which are LCD codes. The dimensions of narrow-sense BCH codes of length $\frac{q^{m} + 1}{q + 1}$ with designed distance $δ = ℓ q^{\frac{m - 1}{2}} + 1$ are determined, where $q > 2$ and $2 \leq ℓ \leq q - 1$ . Lower bounds on the minimum distances of the dual codes of narrow-sense BCH codes of length $q^{m} + 1$ are developed for odd q, which are good in some cases. Moreover, sufficient and necessary conditions for the even-like subcodes of narrow-sense BCH codes of length $q^{m} + 1$ being dually-BCH codes are presented, where q is odd and $m ≢ 0 (\mod 4)$ .

BCH 码是循环码的一个特殊子类，在过去几十年中得到了广泛的研究。然而，确定 BCH 码的参数一直是一个重要但困难的问题。最近，为了进一步研究 BCH 码的对偶码，人们提出了双 BCH 码的概念。本文研究了有限域 Fq 上长度为 qm+1q+1 和 qm+1 的 BCH 码，它们都是 LCD 码。确定了长度为 qm+1q+1 且设计距离为 δ=ℓqm-12+1 的窄义 BCH 码的尺寸，其中 q>2 和 2≤ℓ≤q-1.对于奇数 q，提出了长度为 qm+1 的窄义 BCH 码对偶码的最小距离下限，在某些情况下，下限是很好的。此外，还提出了长度为 qm+1 的窄义 BCH 码的偶样子码成为对偶 BCH 码的充分和必要条件，其中 q 为奇数，m≢0(mod4)。

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引用次数: 0

New results on PcN and APcN polynomials over finite fields 关于有限域上 PcN 和 APcN 多项式的新结果

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-23 DOI: 10.1016/j.ffa.2024.102471

Zhengbang Zha , Lei Hu

Permutation polynomials with low c-differential uniformity have important applications in cryptography and combinatorial design. In this paper, we investigate perfect c-nonlinear (PcN) and almost perfect c-nonlinear (APcN) polynomials over finite fields. Based on some known permutation polynomials, we present several classes of PcN or APcN polynomials by using the Akbary-Ghioca-Wang criterion.

具有低 c 差均匀性的置换多项式在密码学和组合设计中有着重要的应用。本文研究了有限域上的完全 c 非线性（PcN）和近似完全 c 非线性（APcN）多项式。基于一些已知的置换多项式，我们利用 Akbary-Ghioca-Wang 准则提出了几类 PcN 或 APcN 多项式。

引用次数: 0

New results on n-to-1 mappings over finite fields 有限域上 n 对 1 映射的新结果

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-18 DOI: 10.1016/j.ffa.2024.102469

Xiaoer Qin , Li Yan

n-to-1 mappings have many applications in cryptography, finite geometry, coding theory and combinatorial design. In this paper, we first use cyclotomic cosets to construct several kinds of n-to-1 mappings over $F_{q}^{⁎}$ . Then we characterize a new form of AGW-like criterion, and use it to present many classes of n-to-1 polynomials with the form $x^{r} h (x^{q - 1})$ over $F_{q^{2}}^{⁎}$ . Finally, by using monomials on the cosets of a subgroup of $μ_{q + 1}$ and another form of AGW-like criterion, we show some n-to-1 trinomials over $F_{q^{2}}^{⁎}$ .

n 对 1 映射在密码学、有限几何、编码理论和组合设计中有很多应用。在本文中，我们首先利用循环余集构造了几种 Fq⁎ 上的 n 对 1 映射。然后，我们描述了一种新形式的类似 AGW 的准则，并用它提出了许多种在 Fq2⁎上具有 xrh(xq-1) 形式的 n 对 1 多项式。最后，通过使用 μq+1 子群余集上的单项式和另一种形式的类 AGW 准则，我们展示了 Fq2⁎ 上的一些 n 对 1 三项式。

引用次数: 0

Circularity in finite fields and solutions of the equations xm + ym − zm = 1 有限场中的圆周率和方程 xm + ym - zm = 1 的解

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-15 DOI: 10.1016/j.ffa.2024.102467

Wen-Fong Ke , Hubert Kiechle

An explicit formula for the number of solutions of the equation in the title is given when a certain condition, depending only on the exponent and the characteristic of the field, holds. This formula improves the one given by the authors in an earlier paper.

文中给出了当某一条件（仅取决于指数和场的特征）成立时，标题中方程的解的数量的明确公式。这个公式改进了作者在早先论文中给出的公式。

引用次数: 0

Probabilistic Galois theory in function fields 函数域中的概率伽罗瓦理论

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-13 DOI: 10.1016/j.ffa.2024.102466

Alexei Entin, Alexander Popov

We study the irreducibility and Galois group of random polynomials over function fields. We prove that a random polynomial $f = y^{n} + \sum_{i = 0}^{n - 1} a_{i} (x) y^{i} \in F_{q} [x] [y]$ with i.i.d. coefficients $a_{i}$ taking values in the set ${a (x) \in F_{q} [x] : \deg a \leq d}$ with uniform probability, is irreducible with probability tending to $1 - \frac{1}{q^{d}}$ as $n \to \infty$ , where d and q are fixed. We also prove that with the same probability, the Galois group of this random polynomial contains the alternating group $A_{n}$ . Moreover, we prove that if we assume a version of the polynomial Chowla conjecture over $F_{q} [x]$ , then the Galois group of this polynomial is actually equal to the symmetric group $S_{n}$ with probability tending to $1 - \frac{1}{q^{d}}$ . We also study the other possible Galois groups occurring with positive limit probability. Finally, we study the same problems with n fixed and $d \to \infty$ .

我们研究了函数域上随机多项式的不可还原性和伽罗瓦群。我们证明，随机多项式 f=yn+∑i=0n-1ai(x)yi∈Fq[x][y] 的 i.i.d. 系数 ai 以均匀概率在集合 {a(x)∈Fq[x]:dega≤d} 中取值，当 n→∞ 时，d 和 q 是固定的，以趋近于 1-1qd 的概率不可约。我们还证明，以同样的概率，这个随机多项式的伽罗瓦群包含交替群 An。此外，我们还证明，如果我们假设 Fq[x] 上多项式周拉猜想的一个版本，那么这个多项式的伽洛伊群实际上等于对称群 Sn，概率趋于 1-1qd。我们还研究了以正极限概率出现的其他可能的伽罗瓦群。最后，我们研究了 n 固定且 d→∞ 时的相同问题。

引用次数: 0

More classes of permutation pentanomials over finite fields with characteristic two 特性为 2 的有限域上的更多类置换五元数

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-13 DOI: 10.1016/j.ffa.2024.102468

Tongliang Zhang , Lijing Zheng , Hanbing Zhao

<div>Let <math><mi>q</mi><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math>. In this paper, we investigate permutation pentanomials over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math> of the form <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><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> with <math><mrow><mi>gcd</mi></mrow><mo>(</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>t</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>4</mn></mrow></msub></mrow></msup><mo>)</mo><mo>=</mo><mn>1</mn></math>. We transform the problem concerning permutation property of <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> into demonstrating that the corresponding fractional polynomial permutes the unit circle U of <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math> with order <math><mi>q</mi><mo>+</mo><mn>1</mn></math> via a well-known lemma, and then into showing that there are no certain solution in <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></ms

设 q=2m。在本文中，我们研究 Fq2 上形式为 f(x)=xt+xr1(q-1)+t+xr2(q-1)+t+xr3(q-1)+t+xr4(q-1)+t 的置换五次方，gcd(xr4+xr3+xr2+xr1+1,xt+xt-r1+xt-r2+xt-r3+xt-r4)=1。我们将 f(x) 的置换性质问题转化为通过一个著名的 Lemma 证明相应的分数多项式以 q+1 的阶置换 Fq2 的单位圆 U，然后转化为证明与分数多项式相关的 Fq 上的一些高阶方程在 Fq 中没有一定的解。根据数值数据，我们找到了 4≤t<100,1≤ri≤t, i∈[1,4] 的所有此类置换。本文还从单位圆 U 的分数多项式中研究了几种置换多项式。

引用次数: 0

The most symmetric smooth cubic surface over a finite field of characteristic 2 特征为 2 的有限域上最对称的光滑立方曲面

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-13 DOI: 10.1016/j.ffa.2024.102470

Anastasia V. Vikulova

In this paper we find the largest automorphism group of a smooth cubic surface over any finite field of characteristic 2. We prove that if the order of the field is a power of 4, then the automorphism group of maximal order of a smooth cubic surface over this field is ${PSU}_{4} (F_{2})$ . If the order of the field of characteristic 2 is not a power of 4, then we prove that the automorphism group of maximal order of a smooth cubic surface over this field is the symmetric group of degree 6. Moreover, we prove that smooth cubic surfaces with such properties are unique up to isomorphism.

在本文中，我们找到了任意有限特征域 2 上光滑立方曲面的最大自变群。我们证明，如果域的阶是 4 的幂次，那么该域上光滑立方曲面的最大阶自形群是 PSU4(F2)。如果特征 2 场的阶不是 4 的幂次，那么我们证明该场上光滑立方体曲面的最大阶自形群是 6 度对称群。此外，我们还证明了具有这种性质的光滑立方体曲面在同构时是唯一的。

引用次数: 0

Heffter spaces 赫夫特空间

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-08 DOI: 10.1016/j.ffa.2024.102464

M. Buratti , A. Pasotti

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of r mutually orthogonal Heffter systems for any r. Such a set is equivalent to a resolvable partial linear space of degree r whose parallel classes are Heffter systems: this is a new combinatorial design that we call a Heffter space. We present a series of direct constructions of Heffter spaces with odd block size and arbitrarily large degree r obtained with the crucial use of finite fields. Among the applications we establish, in particular, that if $q = 2 k w + 1$ is a prime power with kw odd and $k \geq 3$ , then there are at least $⌈ \frac{w}{4 k^{4}} ⌉$ mutually orthogonal k-cycle systems of order q.

赫夫特阵列的概念在过去十年中备受关注，它等价于一对正交赫夫特系统。在本文中，我们研究了任意 r 的 r 个相互正交的赫夫特系统集合的存在性问题。这样的集合等价于一个度数为 r 的可解析偏线性空间，其并行类是赫夫特系统：这是一种新的组合设计，我们称之为赫夫特空间。我们介绍了一系列直接构造的赫夫特空间，这些空间具有奇数块大小和任意大的度数 r，是利用有限域获得的。在这些应用中，我们特别指出，如果 q=2kw+1 是 kw 为奇数且 k≥3 的素数幂，那么至少有 ⌈w4k4⌉ 个相互正交的 q 阶 k 循环系统。

引用次数: 0

Odd moments for the trace of Frobenius and the Sato–Tate conjecture in arithmetic progressions 算术级数中弗罗贝纽斯迹的奇矩和佐藤塔特猜想

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-08 DOI: 10.1016/j.ffa.2024.102465

Kathrin Bringmann , Ben Kane , Sudhir Pujahari

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2 k + 1)$ -th moment to the zeroeth moment as the size of the finite field $F_{p^{r}}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure.

在本文中，我们考虑了椭圆曲线 Frobenius 的迹的矩，如果迹被限制在一个固定的算术级数上。我们确定了当有限域 Fpr 的大小达到无穷大时，第 (2k+1)-th 矩与第零矩之比的渐近行为。这些结果源于赫维兹类数的和与矩的类似渐近公式，其中和被限制为某些算术级数。作为应用，我们证明了算术级数中弗罗贝尼斯迹的分布与萨托-塔特度量是等分布的。

引用次数: 0

On some congruences and exponential sums 关于一些全等和指数和

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-06-27 DOI: 10.1016/j.ffa.2024.102451

Moubariz Z. Garaev , Igor E. Shparlinski

<div>Let <math><mi>ε</mi><mo>></mo><mn>0</mn></math> be a fixed small constant, <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> be the finite field of p elements for prime p. We consider additive and multiplicative problems in <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let <math><mi>M</mi></math> be an arbitrary subset of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math>. If <math><mi>#</mi><mi>M</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>ε</mi></mrow></msup></math> and <math><mi>H</mi><mo>⩾</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></math> or if <math><mi>#</mi><mi>M</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>5</mn><mo>+</mo><mi>ε</mi></mrow></msup></math> and <math><mi>H</mi><mo>⩾</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>5</mn><mo>+</mo><mi>ε</mi></mrow></msup></math> then all, but <math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>δ</mi></mrow></msup><mo>)</mo></math> elements of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> can be represented in the form hm with <math><mi>h</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>H</mi><mo>]</mo></math> and <math><mi>m</mi><mo>∈</mo><mi>M</mi></math>, where <math><mi>δ</mi><mo>></mo><mn>0</mn></math> depends only on ε. Furthermore, let <math><mi>X</mi></math> be an arbitrary interval of length H and s be a fixed positive integer. If<math><mi>H</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>17</mn><mo>/</mo><mn>35</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>#</mi><mi>M</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>17</mn><mo>/</mo><mn>35</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>,</mo></math> then the number <math><msub><mrow><mi>T</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo></math> of solutions to the congruence<math><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3

让 ε>0 是一个固定的小常数，Fp 是素数 p 的 p 元素有限域。我们考虑 Fp 中涉及区间和任意集合的加法和乘法问题。我们的代表性结果举例如下。设 M 是 Fp 的任意子集。如果 #M>p1/3+ε 和 H⩾p2/3，或者如果 #M>p3/5+ε 和 H⩾p3/5+ε，那么除了 O（p1-δ）个元素外，Fp 的所有元素都可以用 hm 的形式表示，其中 h∈[1,H]，m∈M，δ>0 只取决于 ε。此外，设 X 是长度为 H 的任意区间，s 是一个固定的正整数。若 H>p17/35+ε,#M>p17/35+ε，则全等m1x1s+m2x2s+m3x3s+m4x4s+m5x5s+m6x6s≡λmodp,mi∈M,xi∈X,i=1,...,6 的解的个数 T6(λ)满足T6(λ)=H6(#M)6p(1+O(p-δ))。

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