A THEORY ON SEQUENCE-SPACES AND SHIFT REGISTER GENERATORS

As a means for a unified approach to the description of various shift register generators, the concept of sequence space is first introduced and its properties are examined in various aspects, A sequence space refers to a vector space whose elements are sequences satisfying the r elation specified by a characteristic polynomial, In support of the se quence space, two bases-the elementary basis and the primary basis-are defined, and further, the polynomial expression of sequence is define d as a tool for mathematical manipulations within the sequence space. Based on these definitions, various properties of sequence spaces such as sequence subspaces and minimal sequence spaces are investigated an d summarized in terms of properties and theorems, The developed sequen ce theory is then applied to the description of the behaviors of shift register generators (SRG's). An SRG is represented by the state trans ition matrix, and the relevant SRG sequences are uniquely determined b y this state transition matrix and the initial state vector, For an SR G, it is shown how to identify the sequence space generated by SRG seq uences with a fixed initial state vector (or the SRG space), and furth er, how to find the largest-dimensional sequence space that can be obt ained by varying initial state vectors (or the SRG maximal space), Con versely, for a given sequence space, it is shown how to find the minim um-sized SRG's that can generate the sequence space (or the basic SRG' s), Finally, it is shown that the two typical SRG's-simple SRG and mod ular SRG-are special cases of basic SRG's that can generate the primar y and the elementary bases, respectively.