What is a symmetric channel?

Question

In one of the assignments I was asked to explain whether the given channels are symmetric. For instance, $Q_1$ and $Q_2$, the matrix describes a conditional distribution, e.g., $p(decode_1|encode_1)=p(decode_3|encode_2) =0.5$ $$Q_1 = \begin{bmatrix} 0.5 & 0.3 \\ 0.2 & 0.2 \\ 0.3 & 0.5 \\ \end{bmatrix} $$ ­ $$Q_2 = \begin{bmatrix} 0.5 & 0.2 \\ 0.2 & 0.3 \\ 0.3 & 0.5 \\ \end{bmatrix} $$

I was wondering what is the formal definition of a symmetric channel? Didn't see it in my lecture, tried Wikipedia, only find a topic on binary symmetric channels. Is there a more extended definition of symmetric channels other than binary? ­

Answer 1

According to R.C. Gallager, "Information theory and reliable communication", Wiley (1968):

A DMC is defined to be symmetric, if the set of outputs can be partitioned into subsets in such a way that for each subset the matrix of transition probability has the property that each row is a permutation of each other row and each column is a permutation of each other column.

DMC = discrete memoryless channel.