Tight bounds on the minimum average weight per branch for rate (N-1)/N convolutional codes

Abstract

Consider a cycle in the state diagram of a convolutional code. The average weight per branch of the cycle is equal to the total Hamming weight of all labels on the branches divided by the number of branches. Let w/sub 0/ be the minimum average weight per branch over all cycles in the state diagram, except the zero state self-loop of weight zero. Codes with low w/sub 0/ result in high bit error probabilities when they are used with either Viterbi or sequential decoding. Hemmati and Costello (1980) showed that w/sub 0/ is upper-bounded by 2/sup /spl nu/-2//(3/spl middot/2/sup /spl nu/-2/-1) for a class of (2,1) codes where /spl nu/ denotes the constraint length. In the present correspondence it is shown that the bound is valid for a large class of (n,n-1) codes, n/spl ges/2. Examples of high-rate codes with w/sub 0/ equal to the upper bound are also given. Hemmati and Costello defined a class of codes to be asymptotically catastrophic if w/sub 0/ approaches zero for large /spl nu/. The class of (n,n-1) codes constructed by Wyner and Ash (1963) is shown to be asymptotically catastrophic. All codes in the class have minimum possible w/sub 0/ equal to 1//spl nu/.