Sphere-packing bounds for convolutional codes

We introduce general sphere-packing bounds for convolutional codes. These improve upon the Heller (1968) bound for high-rate convolutional codes. For example, based on the Heller bound, McEliece (1998) suggested that for a rate (n - 1)/n convolutional code of free distance 5 with /spl nu/ memory elements in its minimal encoder it holds that n /spl les/ 2/sup (/spl nu/+1)/2/. A simple corollary of our bounds shows that in this case, n < 2/sup /spl nu//2/, an improvement by a factor of /spl radic/2. The bound can be further strengthened. Note that the resulting bounds are also highly useful for codes of limited bit-oriented trellis complexity. Moreover, the results can be used in a constructive way in the sense that they can be used to facilitate efficient computer search for codes.