American Institute of Mathematical Sciences

In this paper, some general properties of the Zeng-Cai-Tang-Yang cyclotomy are studied. As its applications, two constructions of frequency-hopping sequences (FHSs) and two constructions of FHS sets are presented, where the length of sequences can be any odd integer larger than 3. The FHSs and FHS sets generated by our construction are (near-) optimal with respect to the Lempel–Greenberger bound and Peng–Fan bound, respectively. By choosing appropriate indexes and index sets, a lot of (near-) optimal FHSs and FHS sets can be obtained by our construction. Furthermore, some of them have new parameters which are not covered in the literature.

We investigate subspace codes whose codewords are subspaces of \begin{document}${\rm{PG}}(4,q)$\end{document} having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that \begin{document}$\mathcal{A}_q(5,3) = 2(q^3+1)$\end{document}.

In communication networks theory the concepts of networkness and network surplus have recently been defined. Together with transmission and betweenness centrality, they were based on the assumption of equal communication between vertices. Generalised versions of these four descriptors were presented, taking into account that communication between vertices \begin{document}$ u $\end{document} and \begin{document}$ v $\end{document} is decreasing as the distance between them is increasing. Therefore, we weight the quantity of communication by \begin{document}$ \lambda^{d(u,v)} $\end{document} where \begin{document}$ \lambda \in \left\langle0,1 \right\rangle $\end{document}. Extremal values of these descriptors are analysed.

Let \begin{document}$ R = \mathbb{F}_q[u,v]/\langle u^{2}-1, v^{2}-v, uv-vu\rangle $\end{document} be a finite non-chain ring, where \begin{document}$ q $\end{document} is an odd prime power and \begin{document}$ u^{2} = 1 $\end{document}, \begin{document}$ v^2 = v $\end{document}, \begin{document}$ uv = vu $\end{document}. In this paper, we construct new non-binary quantum codes from (\begin{document}$ \alpha+\beta u+\gamma v+\delta uv $\end{document})-constacyclic codes over \begin{document}$ R $\end{document}. We give the structure of (\begin{document}$ \alpha+\beta u+\gamma v+\delta uv $\end{document})-constacyclic codes over \begin{document}$ R $\end{document} and obtain self-orthogonal codes over \begin{document}$ \mathbb{F}_q $\end{document} by Gray map. By using Calderbank-Shor-Steane (CSS) construction and Hermitian construction from dual-containing (\begin{document}$ \alpha+\beta u+\gamma v+\delta uv $\end{document})-constacyclic codes over \begin{document}$ R $\end{document}, some new non-binary quantum codes are obtained.

At Eurocrypt 2015, Barbulescu et al. introduced two new methods of polynomial selection, namely the Conjugation and the Generalised Joux-Lercier methods, for the number field sieve (NFS) algorithm as applied to the discrete logarithm problem over finite fields. A sequence of subsequent works have developed and applied these methods to the multiple and the (extended) tower number field sieve algorithms. This line of work has led to new asymptotic complexities for various cases of the discrete logarithm problem over finite fields. The current work presents a unified polynomial selection method which we call Algorithm \begin{document}$ \mathcal{D} $\end{document}. Starting from the Barbulescu et al. paper, all the subsequent polynomial selection methods can be seen as special cases of Algorithm \begin{document}$ \mathcal{D} $\end{document}. Moreover, for the extended tower number field sieve (exTNFS) and the multiple extended TNFS (MexTNFS), there are finite fields for which using the polynomials selected by Algorithm \begin{document}$ \mathcal{D} $\end{document} provides the best asymptotic complexity. Suppose \begin{document}$ Q = p^n $\end{document} for a prime \begin{document}$ p $\end{document} and further suppose that \begin{document}$ n = \eta\kappa $\end{document} such that there is a \begin{document}$ c_{\theta}>0 $\end{document} for which \begin{document}$ p^{\eta} = L_Q(2/3, c_{\theta}) $\end{document}. For \begin{document}$ c_{\theta}>3.39 $\end{document}, the complexity of exTNFS-\begin{document}$ \mathcal{D} $\end{document} is lower than the complexities of all previous algorithms; for \begin{document}$ c_{\theta}\notin (0, 1.12)\cup[1.45, 3.15] $\end{document}, the complexity of MexTNFS-\begin{document}$ \mathcal{D} $\end{document} is lower than that of all previous methods.

We show that there is a binary subspace code of constant dimension 3 in ambient dimension 7, having minimum subspace distance 4 and cardinality 333, i.e., \begin{document}$ 333 \le A_2(7, 4;3) $\end{document}, which improves the previous best known lower bound of 329. Moreover, if a code with these parameters has at least 333 elements, its automorphism group is in one of 31 conjugacy classes.

This is achieved by a more general technique for an exhaustive search in a finite group that does not depend on the enumeration of all subgroups.

There are two standard approaches to the construction of \begin{document}$ t $\end{document}-designs. The first one is based on permutation group actions on certain base blocks. The second one is based on coding theory. The objective of this paper is to give a spectral characterisation of all \begin{document}$ t $\end{document}-designs by introducing a characteristic Boolean function of a \begin{document}$ t $\end{document}-design. The spectra of the characteristic functions of \begin{document}$ (n-2)/2 $\end{document}-\begin{document}$ (n, n/2, 1) $\end{document} Steiner systems are determined and properties of such designs are proved. Delsarte's characterisations of orthogonal arrays and \begin{document}$ t $\end{document}-designs, which are two special cases of Delsarte's characterisation of \begin{document}$ T $\end{document}-designs in association schemes, are slightly extended into two spectral characterisations. Another characterisation of \begin{document}$ t $\end{document}-designs by Delsarte and Seidel is also extended into a spectral one. These spectral characterisations are then compared with the new spectral characterisation of this paper.

In this paper, by analyzing the quadratic factors of an \begin{document}$ 11 $\end{document}-th degree polynomial over the finite field \begin{document}$ {\mathbb F}_{2^n} $\end{document}, a conjecture on permutation trinomials over \begin{document}$ {\mathbb F}_{2^n}[x] $\end{document} proposed very recently by Deng and Zheng is settled, where \begin{document}$ n = 2m $\end{document} and \begin{document}$ m $\end{document} is a positive integer with \begin{document}$ \gcd(m,5) = 1 $\end{document}.

We show that under certain conditions every maximal symmetric subfield of a central division algebra with positive unitary involution \begin{document}$ (B, \tau) $\end{document} will be a Galois extension of the fixed field of \begin{document}$ \tau $\end{document} and will "real split" \begin{document}$ (B, \tau) $\end{document}. As an application we show that a sufficient condition for the existence of positive involutions on certain crossed product division algebras over number fields, considered by Berhuy in the context of unitary space-time coding, is also necessary, proving that Berhuy's construction is optimal.