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 ν memory elements in its minimal encoder it holds that n ≤ 2^(ν+1)2/. A simple corollary of our bounds shows that in this case, n < 2^ν2/, an improvement by a factor of √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.