Research On The Depth Distribution Of Linear Codes And The Generalized Derivatives Of Binary Sequences

The derivative of sequences is an important tool to depict and analyze the complexity of sequences and the depth distribution of linear codes.The application of derivative to the field of encoding and stream cipher is studied,the theory of the depth distribution in finite fields and finite rings are extended.The generalized derivatives of binary multisequences are creatively defined.Some properties and structures of the generalized derivatives of periodic multisequences are discussed.The main researches of this dissertation are as follows.1)The width of codewords over the finite ring F_p+uF_p is defined,some properties of the width and a recursive algorithm for computing the width of codewords over the finite ring F_p +uF_p are given.2)The width of codewords over the finite ring Z_p^kis defined,some properties of the width and two recursive algorithms for computing the width of codewords over the finite ring Z_p^kare given.This research generalizes the depth distribution theories and enriches the encoding theory over finite rings.3)The period and the generalized derivatives of binary multisequences are defined.The generalized derivatives of periodic binary multisequences are proved to be periodic binary multisequences.The connection between the period of the generalized derivatives of periodic binary multisequences and that of themselves is studied,then the connection between the structure of multisequences with period 2^N and 2^N-land the structure of themselves is also researched.