Finite Fields and Their Applications最新文献

In this paper we compute the automorphism group of the curves $X_{a,b,n,s}$ and $Y_{n,s}$ introduced in Tafazolian et al. [27] as new examples of maximal curves which cannot be covered by the Hermitian curve. They arise as subcovers of the first generalized GK curve (GGS curve). As a result, a new characterization of the GK curve, as a member of this family, is obtained.

Blokhuis showed that all maximum cliques in Paley graphs of square order have a subfield structure. Recently, it has been shown that in Peisert-type graphs, all maximum cliques are affine subspaces, and yet some maximum cliques do not arise from a subfield. In this paper, we investigate the existence of a clique of size $\sqrt{q}$ with a subspace structure in pseudo-Paley graphs of order q from unions of semi-primitive cyclotomic classes. We show that such a clique must have an equal contribution from each cyclotomic class and that most such pseudo-Paley graphs do not admit such cliques, suggesting that the Delsarte bound $\sqrt{q}$ on the clique number can be improved in general. We also prove that generalized Peisert graphs are not isomorphic to Paley graphs or Peisert graphs, confirming a conjecture of Mullin.

It is known that a Bruen chain of the three-dimensional projective space $\mathrm{PG}(3,q)$ exists for every odd prime power q at most 37, except for $q=29$. It was shown by Cardinali et al. (2005) that Bruen chains do not exist for $41⩽q⩽49$. We develop a model, based on finite fields, which allows us to extend this result to $41⩽q⩽97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$. Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs.

A complete complementary code (CCC) consists of M sequence sets with size M. The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results.

The classification of the 2-designs with $\lambda =2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in [2]. The result is achieved by using conics and hyperovals of $PG(2,q)$.

In this paper, we generalize [6], [1], [5] and [3] by allowing the distance between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of $F_q^d$ for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from $Q\left(\sqrt{-23}\right)$ and $Q\left(\sqrt{-31}\right)$, which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.

Let q be a prime power. For $m=2,3,4$, we construct stable polynomials of the form $b^{m-1}{(x+a)}^m+c(x+a)+d$ over $F_q$ by Capelli's lemma. Moreover, when $m=2$ and $q\equiv 1\left(\mathrm{mod}4\right)$, we improve a lower bound for the number of stable quadratic polynomials over $F_q$ due to Goméz-Pérez and Nicolás [4]. When $m=3$, we prove Ahmadi and Monsef-Shokri's conjecture [2] that $x^3+x^2+1$ is stable over $F_2$.

Let q be a power of a prime such that $q\equiv 1\left(\mathrm{mod}r\right)$. Let c be an r-th power residue over $F_q$. In this paper, we present a new r-th root formula which generalizes G.H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. Our algorithm which is based on the recurrence relations arising from irreducible polynomial $h\left(x\right)=x^r+{(-1)}^{r+1}(b+{(-1)}^{r}r)(x-1)$ with $b=c+{(-1)}^{r+1}r$ requires only $O(r^2\log q+r^4)$ multiplications for $r>1$. The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms.

This paper defines a linear representation for nonlinear maps $F:F^n\to F^n$ where $F$ is a finite field, in terms of matrices over $F$. This linear representation of the map F associates a unique number N and a unique matrix M in $F^{N\times N}$, called the Linear Complexity and the Linear Representation of F respectively, and shows that the compositional powers $F^{\left(k\right)}$ are represented by matrix powers $M^k$. It is shown that for a permutation map F with representation M, the inverse map has the linear representation $M^{-1}$. This framework of representation is extended to a parameterized family of maps $F_{\lambda }\left(x\right):F\to F$, defined in terms of a parameter $\lambda \in F$, leading to the definition of an analogous linear complexity of the map $F_{\lambda }\left(x\right)$, and a parameter-dependent matrix representation $M_{\lambda }$ defined over the univariate polynomial ring $F\left[\lambda \right]$. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation $M_{\lambda }$. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map F, and to the group generated by a finite number of permutation maps over $F$.