In this paper, we prove that a binary sequence is perfect (resp., quasi-perfect) if and only if its support set for any flnite group (not necessarily cyclic) is a Hadamard difference set of type Ⅰ (resp., type Ⅱ); and we prove that the kernel of any nonzero linear functional (or the image of any linear transformation A with dim (KerA)=1) on the linear space GF (2m) over the fleld GF (2) (excluding 0) is a cyclic Hadamard difference set of type II using Gaussian sums; and we prove that the multiplier group of the above difference set is equal to the Galois group Gal (GF (2m)=GF (2)); and we mention the relationship between the Hadamard transform and the irreducible complex characters

Introduced in this note are some new methods of constructing perfect arrays. Some results on the liftability of the perfect ternary sequences are obtained. In addition, a perfect ternary array with index group Zr2 is presented.

周期拟完美序列（即，周期自相关函数的旁瓣值都为-1的二元序列）在CDMA移动通信和流密码中都有重要应用。本文提示了周期拟完美序列与循环Hadamard（Ⅱ）型差集的深刻对应关系，进而建立了周期拟完美序列的计数公式；并且通过决定循环Hadamar（Ⅱ）型差集的乘子群，分别求了各族循环Hadamard（Ⅱ）型差集对应的本质不同的拟完美序列的个数。

It is already known that there are several nonlinearity criteria such as algebraic degree, nonlinearity, distance to linear structures, correlation im-mune, propagation criterion, differential uniformity, which are used to check whether a cryptographic function is weak or not. In this paper we will dis-cuss these criteria from a valuation point of view, and consider the largest transformation group which leave a criterion invariant, which is named its symmetry group. It can serve as a way of comparing the stability of nonlin-earity criteria under the action of invertible transformations.

本文完善了我们在1992年以来提出的研究乘子猜想的特征标方法，从而对n=3n1情形的乘子猜想取得了较大的进展。概略地说，我们证明了：n=3nr的情形，用（n1，λ）=1代替n1>λ，第二乘子定理仍然成立。进而我们证明了：在n=3pr的情形，把p>λ的条件去掉，第一乘子定量仍然成立。即，设D是abel群G的一个（υ，к，λ）一差集，n=3pr，p是素数，且（p，υ）=1。则P是D的数值乘子。