A state space approach to periodic convolutional codes

In this paper we study periodically time-varying convolutional codes by means of input-state-output representations. Using these representations we investigate under which conditions a given time-invariant convolutional code can be transformed into an equivalent periodic time-varying one. The relation between these two classes of convolutional codes is studied for period 2. We illustrate the ideas presented in this paper by constructing a periodic time-varying convolutional code from a time-invariant one. The resulting periodic code has larger free distance than any time-invariant convolutional code with equivalent parameters

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

[EN] 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.S

State-Space Realizations of Periodic Convolutional Codes

Convolutional codes are discrete linear systems over a finite field and can be defined as F[d]-modules, where F[d] is the ring of polynomials with coefficient in a finite field F. In this paper we study the algebraic properties of periodic convolutional codes of period 2 and their representation by means of input-state-output representations. We show that they can be described as F[d2]-modules and present explicit representation of the set of equivalent encoders. We investigate their state-space representation and present two different but equivalent types of state-space realizations for these codes. These novel representations can be implemented by realizing two linear time-invariant systems separately and switching the input (or the output) that is entering (or leaving) the system. We investigate their minimality and provide necessary and also sufficient conditions in terms of the reachability and observability properties of the two linear systems involved. The ideas presented here can be easily generalized for codes with period larger than 2.This work was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (FCT-Fundaçao para a Ciência e a Tecnologia) within project UIDB/04106/2020. It was also partially supported by Base Funding (UIDB/00147/2020) and Programmatic Funding (UIDP/00147/2020) of the Systems and Technologies Center - SYSTEC - funded by national funds through the FCT/MCTES (PIDDAC). The work of the second author was partially supported by Spanish grants PID2019-108668GB-I00 of the Ministerio de Ciencia e Innovación of the Gobierno de España and VIGROB-287 of the Universitat d'Alacant