The normalized second moment of the binary lattice determined by a convolutional code

Calculates the per-dimension mean squared error /spl mu/(S) of the two-state convolutional code C with generator matrix /spl lsqb/1,1+D/spl rsqb/, for the symmetric binary source S=(0,1), and for the uniform source S=/spl lcub/0,1/spl rcub/. When S=(0,1), the quantity /spl mu/(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a random binary sequence from the code. When S=/spl lcub/0,1/spl rcub/, the quantity /spl mu/(S) is the second moment of the Voronoi region of the module 2 binary lattice determined by C. The key observation is that a convolutional code with 2/sup /spl upsi// states gives 2/sup /spl upsi// approximations to a given source sequence, and these approximations do not differ very much. It is possible to calculate the steady state distribution for the differences in these path metrics, and hence, the second moment. The authors only give details for the convolutional code /spl lsqb/1,1+D/spl rsqb/, but the method applies to arbitrary codes. They also define the covering radius of a convolutional code, and calculate this quantity for the code /spl lsqb/1,1+D/spl rsqb/.< >