On the Enumeration and Generation of Nonweight Equivalent Rate 1/2 Convolutional Codes

Abstract

The weight equivalence of rate $\frac{1}{2}$ binary convolutional codes and the problem of enumeration and generation of all nonweight equivalent codes is considered for a given memory order m. This is done during the generation process by first ordering the encoders into classes such that it becomes easy to recognize the weight equivalence as well as the catastrophic error propagation conditions. Subclasses of respectively self and nonself reciprocal as well as catastrophic and noncatastrophic encoders are introduced. The cardinal number of these classes and subsequently the number of “nonweight equivalent” codes are then computed recursively as a function of m. Finally, since all the relations amount to simple convolutions they are compactly represented by generating functions which are tabulated.