Some properties of binary convolutional code generators

This paper formalizes the discussion of some structural properties of the generators for binary convolutional codes. The use of these properties may be helpful in the selection of generators which produce codes with desired error-correcting properties for sequential decoding. The approach taken is to decompose a generator sequence into subsequences called "subgenerators." The set of all such possible subsequences starting with a 1 is shown to form an Abelian group with respect to a binary convolution. The recurrence relation which permits the construction of the inverse of an encoding subgenerator is given. One application of these results is a simple proof of the reproducing property of the truncated convolutional message set noted by Wozencraft and Reiffen. The notion of "adjoint" canonical generators, all of which have the same error-correcting properties but different message sets, is also introduced. The distinction between encoding and decoding constraint lengths is pointed out and an estimate made of the achievable difference between the two. An efficient search procedure to select the generator of rate 1/2 which possesses the best error-correcting properties is also discussed. Selected generators of rates 1/2 and 1/3 are tabulated up to (32, 16) and (21, 7) , respectively.