On convolutional codes and sphere packing bounds

We introduce general sphere packing bounds for convolutional codes. These improve upon the Heller bound [J.L.Heller (1968)] for high rate convolutional codes. For example, based on the Heller bound, McEliece [2, p. 1114] suggested that for a rate (n-1)/n code of free distance 5 with ν memory elements in its minimum encoder, asymptotically as ν → ∞ 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.