Symmetries of the State Diagram of the Syndrome Former of a Binary Rate-1/2 Convolutional Code

Abstract

We extend and generalize some earlier results [1] on syndrome decoding of binary rate-1/2 convolutional codes. Figure 1 represents a familiar example of such a code. The additions in Figure 1 are modulo-2 and all binary sequences ...,b₋₁,b₀, b₁,... are represented as power series b(α) = ... + b₋₁α^-1 + b₀ + b₁α + .... The encoder has connection polynomials C₁(α) = 1 + α² and C₂(α) = 1 + α + α².