Concordant and partially concordant ring convolutional codes for CPFSK

We consider new M-ary convolutional codes over the ring of integers Z_M for full response continuous phase frequency shift keying (CPFSK) with modulation index h=p/g, where positive integers p and q have no common factors. In comparison with the other best known constructions of encoded CPFSK, our encoder is feedback-free. The new trellis codes are designed for additive white Gaussian noise channels, where in the receiver we carry out Viterbi decoding on the joint trellis of encoded CPFSK. In such a case the decoding complexity is mainly defined by the number of states of joint trellis S, and S_c⩽S⩽S_cq, where S_c is the number of states in the trellis diagram of the convolutional code and q defines the number of states in the trellis diagram of CPFSK. We show, that there exist convolutional codes for which S=S_cq/GCD(M,q). It follows that if GCD(M,q)=q then S=S_c . We call such convolutional codes concordant codes. Then if GCD(M,q)=1 we have S=S_cq and such codes are non-concordant codes. Finally if 1<GCD(M,q)<q then S_c<S<S(_c q) and we call such convolutional codes partially concordant codes