@article {24227,
	title = {  Tight bounds on the minimum average weight per branch for rate (N-1)/N convolutional codes},
	journal = {IEEE Transactions on Information Theory },
	volume = {43},
	year = {1997},
	pages = {1301-1305},
	publisher = {IEEE},
	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 w0\&nbsp;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\&nbsp;result in high bit error probabilities when they are used with either Viterbi or sequential decoding. Hemmati and Costello (1980) showed that w0\&nbsp;is upper-bounded by 2ν-2/(3{\textperiodcentered}2ν-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\&nbsp;equal to the upper bound are also given. Hemmati and Costello defined a class of codes to be asymptotically catastrophic if w\&nbsp;0\&nbsp;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\&nbsp;equal to 1/ν},
	keywords = {Convolutional codes, error statistics, minimisation},
	doi = {10.1109/18.605599},
	author = {Hole, M.F. and Kjell J{\o}rgen {Hole}}
}
