Linear Multivalued Sequential Coding Networks

Linear multivalued sequential coding networks are circuits whose input and output are synchronized sequences of nonnegative integers less than some fixed numberm. The output depends linearly on the present input and a finite number of previous inputs and outputs. The transfer characteristics of such a network are described by a ratio of polynomials in the delay operator, where the multiplication and addition are performed with respect to the fixed modulusm. An algebraic theory of the delay polynomials is obtained. It is shown that a polynomial has a complete set of null sequences if, and only if, its first and last coefficients are prime to the modulus m. The polynomials with no null sequences are characterized. It is shown when common null sequences imply that the polynomials have common factors and that a complete set of null sequences defines the polynomial. It is also shown that a transfer function can be realized if the denominator contains a constant term prime tomand explicit constructions are given. A network is stable if the polynomial in the denominator of the transfer junction has no null sequence. Thus any nontrivial polynomial or its inverse is unstable if we are working moduloaprime. If the modulus is not prime, stable networks with stable inverses are constructed. Finally it is indicated how polynomials with no null sequences can be used to simplify the construction of coding networks.