Advances in Mathematics of Communications最新文献

In 2002 Mauduit and Sárközy started to study finite sequences of begin{document}$ k $end{document} symbols

begin{document}$ E_{N} = left(e_{1},e_{2},cdots,e_{N}right)in mathcal{A}^{N}, $end{document}

where begin{document}$ mathcal{A} = left{a_{1},a_{2},cdots,a_{k}right}(kin mathbb{N},kgeq 2) $end{document} is a finite set of begin{document}$ k $end{document} symbols. Later many pseudorandom sequences of begin{document}$ k $end{document} symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the begin{document}$ k $end{document}-ary Sidel'nikov sequences with length begin{document}$ q-1 $end{document} by using the estimates for certain character sums with exponential function, where begin{document}$ q $end{document} is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size begin{document}$ phi(q-1) $end{document} of begin{document}$ k $end{document}-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that begin{document}$ k = o(q^{frac{1}{4}}) $end{document}, where begin{document}$ phi $end{document} denotes Euler's totient function.

Let begin{document}$ mathbb{F}_{q} $end{document} be a finite field with character begin{document}$ p $end{document}. In this paper, the multiplicative group begin{document}$ mathbb{F}_{q}^{*} = mathbb{F}_{q}setminus{0} $end{document} is decomposed into a mutually disjoint union of begin{document}$ gcd(6l^mp^n,q-1) $end{document} cosets over subgroup begin{document}$ $end{document}, where begin{document}$ xi $end{document} is a primitive element of begin{document}$ mathbb{F}_{q} $end{document}. Based on the decomposition, the structure of constacyclic codes of length begin{document}$ 6l^mp^n $end{document} over finite field begin{document}$ mathbb{F}_{q} $end{document} and their duals is established in terms of their generator polynomials, where begin{document}$ pneq{3} $end{document} and begin{document}$ lneq{3} $end{document} are distinct odd primes, begin{document}$ m $end{document} and begin{document}$ n $end{document} are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length begin{document}$ 6l^mp^n $end{document} over begin{document}$ mathbb{F}_{q} $end{document}.

One of the fundamental problems in coding theory is to find begin{document}$ n_q(k,d) $end{document}, the minimum length begin{document}$ n $end{document} for which a linear code of length begin{document}$ n $end{document}, dimension begin{document}$ k $end{document}, and the minimum weight begin{document}$ d $end{document} over the field of order begin{document}$ q $end{document} exists. The problem of determining the values of begin{document}$ n_q(k,d) $end{document} is known as the optimal linear codes problem. Using the geometric methods through projective geometry and a new extension theorem given by Kanda (2020), we determine begin{document}$ n_3(6,d) $end{document} for some values of begin{document}$ d $end{document} by proving the nonexistence of linear codes with certain parameters.

Let begin{document}$ G_2(q) $end{document} be a Chevalley group of type begin{document}$ G_2 $end{document} over a finite field begin{document}$ mathbb{F}_q $end{document}. Considering the begin{document}$ G_2(q) $end{document}-primitive action of rank begin{document}$ 3 $end{document} on the set of begin{document}$ frac{q^3(q^3-1)}{2} $end{document} hyperplanes of type begin{document}$ O_{6}^{-}(q) $end{document} in the begin{document}$ 7 $end{document}-dimensional orthogonal space begin{document}$ {{rm{PG}}}(7, q) $end{document}, we study the designs, codes, and some related geometric structures. We obtained the main parameters of the codes, the full automorphism groups of these structures, and geometric descriptions of the classes of minimum weight codewords.