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A [[Communication channel|communication channel]] whose transition function possesses some kind of symmetry. A homogeneous discrete time [[Memoryless channel|memoryless channel]] with finite alphabets  $  Y $
and  $  \widetilde{Y}  = Y $
of input and output letters, respectively, and defined by a matrix of transition probabilities  $  \{ q ( y, \widetilde{y}  ) :  y, \widetilde{y}  \in Y \} $
is called a symmetric channel if:

$$ \tag{* }
q ( y, \widetilde{y}  )  = \ 
\left \{

\begin{array}{ll}
 q  & \textrm{ when }  y = \widetilde{y}  ,  \\

\frac{1 - q }{n - 1 }
  & \textrm{ when }  y \neq \widetilde{y}  ,  \\
\end{array}

 \right .$$

where  $  n $
is the number of elements of  $  Y $, 
$  0 \leq  q \leq  1 $. 
The most studied example of a memoryless symmetric channel is the binary symmetric channel with matrix of transition probabilities

$$ 
\left \| 
\begin{array}{cc}
 q  &1 - q  \\
1 - q  & q  \\
\end{array}
 \ 
\right \| .
$$

For symmetric channels, many important information-theoretic characteristics can either be calculated explicitly or their calculation can be substantially simplified in comparison with non-symmetric channels. For example, for a memoryless symmetric channel with matrix  $  \{ q ( y, \widetilde{y}  ) :  y, \widetilde{y}  \in Y \} $
of the form (*) the capacity  $  C $(
cf. [[Transmission rate of a channel|Transmission rate of a channel]]) is given by the equation

$$ 
C  =   \mathop{\rm log}  n + q   \mathop{\rm log}  q + ( 1 - q)  \mathop{\rm log}  
\frac{1 - q }{n - 1 }
 .
$$

For references see ,

cited under [[Communication channel|Communication channel]].

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.C. Gallager,   "Information theory and reliable communication" , Wiley  (1968)</TD></TR></table>