Sequential Decoding: Computational Complexity and the Cutoff Rate

Sequential decoding algorithms decode convolutional codes by guessing their way through the expanding tree of possible transmitted sequences. In this way computational complexity is reduced. Generally it comes at the cost of having to communicate at rates distinctly below capacity. The computational cutoff rate Rcomp delineates the border between those rates for which the average decoding time is bounded and those for which it is infinite. A series of papers establishes Rcomp = E0(1), where E0(ρ) is the Gallager function. 1 Convolutional Codes Convolutional codes use memory to spread the message digits over time in order to average out the channel noise. An example of a convolutional encoder is shown in figure (1). The encoder is defined by three parameters λ -the number of digits inputed at each time point, L -the constraint length and ν -the number of digits outputed at each time point. The rate of the code is defined as R , λ/ν log(2) in nats. The constraint length plays the same role as the length of a block code. If the sequence of message digits is denoted u (1) 1 , u (2) 1 , . . . , u (λ) 1 , u (1) 2 , u (2) 2 , . . . , u (λ) 2 , . . . then the sequence of channel digits outputed by the convolutional encoder x (1) 1 , x (2) 1 , . . . , x (ν) 1 , x (1) 2 , x (2) 2 , . . . , x (ν) 2 , . . .