On the State Approach Representations of Convolutional Codes over Rings of Modular Integers

Abstract

In this study, we prove the existence of minimal first-order representations for convolutional codes with the predictable degree property over principal ideal artinian rings. Further, we prove that any such first-order representation leads to an input/state/output representation of the code provided the base ring is local. When the base ring is a finite field, we recover the classical construction, studied in depth by J. Rosenthal and E. V. York. This allows us to construct observable convolutional codes over such rings in the same way as is carried out in classical convolutional coding theory. Furthermore, we prove the minimality of the obtained representations. This completes the study of the existence of input/state/output representations of convolutional codes over rings of modular integers.

Natural Factorization of Linear Control Systems through Parallel Gathering of Simple Systems

“…In particular, the algebraic study of linear systems in the state-space approach [3] deals with linear systems defned on algebras and modules over a commutative ring [4,5]. Tis approach has been used recently in the feld of convolutional codes [6][7][8][9][10]. Convolutional codes are in fact error-correcting codes over a fnite feld F defned as vector subspaces of F(z) n , where F (z) is the feld of rationals which are realized as linear control systems over F .…”