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

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 w₀ 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 w₀ result in high bit error probabilities when they are used with either Viterbi or sequential decoding. Hemmati and Costello (1980) showed that w₀ is upper-bounded by 2^ν-2/(3·2^ν-2-1) for a class of (2,1) codes where ν 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⩾2. Examples of high-rate codes with w₀ equal to the upper bound are also given. Hemmati and Costello defined a class of codes to be asymptotically catastrophic if w ₀ approaches zero for large ν. 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 w₀ equal to 1/ν