Asymptotically catastrophic convolutional codes

Abstract

The minimum distance growth rate of unmerged codewords in a convolutional code is shown to depend upon the minimum average weight per branch <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w_{0}</tex> in the encoder state diagram. An upper bound on <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w_{0}</tex> is obtained for a large class of rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> codes which includes many of the best known classes of rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> codes. The hound is shown to be tight for short constraint length codes. A class of codes is defined to be asymptotically catastrophic if <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">w_{0}</tex> approaches zero for large constraint lengths. Several classes of rate <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</tex> codes are shown to be asymptotically catastrophic. These include classes containing codes known to have large free distance. It is argued that the free distance alone is not a sufficient criterion to determine a codes performance with either Viterbi or sequential decoding. A code with a low distance growth rate will yield a high bit error probability and will not perform well with truncated Viterbi decoding.