格雷码的代数结构和分形生成的递归算法

利用矩阵理论讨论了格雷码的代数结构。文中给出的定理揭示了格雷码与自然码之间的联系，格雷变换对方幂分组的封闭性及格雷变换的整体周期性。定理的证明基于如下递归过程：Ｅ２＾ｎ＝［Ｅ２＾ｎ－１ ０ ０ Ｅ↑￣２＾ｎ－１］，Ｅ↑￣２＾ｎ＝［０ Ｅ２＾ｎ－１ Ｅ↑￣２＾ｎ－１ ０］，ｎ∈Ｚ＾＋，Ｅ１＝［１］，Ｅ↑￣＝ ［１］提出一类新的广义Ｋｒｏｎｅｃｋｅｒ乘积，其中矩阵元素的乘法定义为ａ＊ｂ＝２ｍａ＋ｂ。按

The Algebraic Structure of Gray Code and the Recursive Algorithms for Generating Fractals

An algebraic structure of Gray code is discussed by using matrix theory.The theorems show the relations between Gray code and naturel code, the closedness of Gray transformation under the group by power 2 and the global periodicity of the transformation. The proof of the theorems mentioned above is based on the following recursive procedure:The paper proposes a new class of generalized Kronecker product, where the product for two elements of matrix is defined by a * b = 2ma + b from which six kinds of the recursive algorithms are given. These algorithms are simple and efficient in generating fractals. The paper also investigates the relations beween the recursive algorithms and iterative function system.