THE NORMALIZED 2ND MOMENT OF THE BINARY LATTICE DETERMINED BY A CONVOLUTIONAL CODE

We calculate the per-dimension mean squarred error mu(S) of the two-st ate convolutional code C with generator matrix [1,1 + D], for the symm etric binary source S = {0, 1}, and for the uniform source S = [0, 1]. When S = {0, 1}, the quantity mu(S) is the second moment of the coset weight distribution, which gives the expected Hamming distance of a r andom binary sequence from the code. When S = [0, 1], the quantity mu( S) is the second moment of the Voronoi region of the modulo 2 binary l attice determined by C. The key observation is that a convolutional co de with 2v states gives 2v approximations to a given source sequence, and these approximations do not differ very much. It is possible to ca lculate the steady state distribution for the differences in these pat h metrics, and hence, the second moment. In this paper we shall only g ive details for the convolutional code [1; 1 + D], but the method appl ies to arbitrary codes. We also define the covering radius of a convol utional code, and calculate this quantity for the code [1, 1 + D].