Subspaces of GF(q)^w and Convolutional Codes

In this paper we demonstrate that among all subspaces of GF(q) ! convolutional codes are best suited for error control purposes. To this end we regard several deening properties of convolutional codes and study the classes of subspaces deened by each of those properties alone. It turns out that these superclasses of the class of convolutional codes either achieve no better distance to rate ratio or are susceptible to an unavoidable innnite error propagation. We consider convolutional codes as subspaces of the space GF(q) ! of all seminiinite words over GF(q) provided with the metric deened as follows (;) = (0 , if = ; maxf1=n : (n) 6 = (n)g , if 6 = : A subspace L of GF(q) ! is a linear subset of GF(q) ! which is closed in the topology deened by the metric. Following Forney F1] a convolutional encoder is a k-input n-output time-invariant nite-state linear sequential circuit. All these properties are in some sense transferred to the set of all seminiinite output sequences , that is, to the convolutional code. Another important property of convolutional codes is an eeect which Forney F2] calls remerging to the all zero path. If a subspace L of GF(q) ! does not possess this property, any error-correcting decoder for L is susceptible to an unavoidable innnite error propagation, that is, decoding errors in a nite initial part of the received sequence can force the decoder to make further on an innnite number of decoding errors. We list these properties below using the following notations: