Correlated Decoding for Channels with Arbitrarily Varying Channel Probability Functions

Let X = I1, " '" , a}, Y = {1, ' " , b} be finite sets. A stochastic matrix w with a rows and b columns will be called a channel. X, Y are the input and output alphabets (respectively) of the channel. We denote the set of all channels with input alphabet X and output alphabet Y by e (X, Y). A channel w E C (X, Y) can be used for communication from one person, the sender, to another person, the receiver. There is given in advance a finite set of messages ~ = {1, ., N}, one of which will be presented to the sender for transmission. We allow the sender a randomized encoding and the receiver a randomized decoding (cf. [4], [5]). ~[ore precisely, the sender encodes the message by an encoding channel E E e ( ~ , X) with E(v, x) being the probability tha t input x is given to channel w when message ~ is presented to the sender for transmission. When the receiver observes the output y of the transmission channel w, he decodes it by a decoding channel D E C( Y, !~) with D(y, ~) being the probability tha t the receiver will decide tha t message ~ is intended. The matrix e = e(E, D, w) = E . w . D E e ( ~ , 3 ) is the error matrix of code (E, D) for channel w. I ts element e@, ~) gives the probability that, when ~ is presented to the sender the receiver will decide that message/~ is intended, when code (E, D) is used on channel w. The average error probability over all messages in the set ~ is therefore