It is well known that a convolutional code is essentially a linear sys
tem defined over a finite field, In this paper we elaborate on this co
nnection. We will define a convolutional code as the dual of a complet
e linear behavior in the sense of Willems. Using ideas from systems th
eory, we describe a set of generalized first-order descriptions for co
nvolutional codes. As an application of these ideas, we present a new
algebraic construction for convolutional codes.