From the Introduction: “Let \(A\) and \(B\) be two finite sets. \(A\) is the input alphabet and \(B\) is the output one. A channel from \(A\) to \(B\) is any transition probability from \(A\) to \(B\). […]
“There is a finite set \(M\) of messages. In order to be sent using the channel \(W\) they are firstly encoded and then decoded. There are many ways to define a code. For us, a code is a pair \((f,\varphi)\), where \(f\colon M\to A^n\) is a one-to-one function (the encoder) and \(\varphi\colon B^n\to M\) is the decoder defined by the maximum likelihood rule […]
“Any channel has an error. We shall deal with the average error, that is, \[ E_f(W^{\otimes n}):=\frac{1}{| m|} \sum_{m\in M}\text{Pr}(\varphi(f(m))\neq m).{(1.5)} \] […]
“By the best encoder \(f\) we understand that one for which \(E_f\) is minimum. In that case the error depends only on \(n\) and \(W\). That is why we shall be interested in the quantity \[ E_n(W):=\min\{E_f(W^{\otimes n})\mid f\colon M\to A^n\}.{(1.8)} \] […]
“Our goal is to investigate the minimum number of trials in order to find \(E_{n,k}(W)\), where \[ E_{n,k}(W):=\min\{E_n(W)\mid | M|=k\}.\text{''} \]