A DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ is called symmetric if the rows of $Q$ are permutations of each other and the columns of $Q$ are permutations of each other

A DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$ is called weakly symmetric if the rows of $Q$ are permutations of each other and the column sums in $Q$ are equal i.e. $$\sum\limits_{a\in\mathcal{X}}P_{Y|X}(b|a)=c \ \mbox{(constant), }\forall b\in \mathcal{Y}$$

A DMC $(\mathcal{X},\mathcal{Y},Q=[P_{XY}])$, called quasi-symmetric if $Q$ can be partitioned along its columns into $m$ weakly symmetric sub-matrices $Q_{1},\cdots, Q_{m}$ for some integer $m\ge1$, where each sub-matrix $Q_{i}$ has size $|\mathcal{X}|\times|\mathcal{Y}_{i}|$, $i=1,\cdots,m$, with $$\mathcal{Y}_{1}\cup\cdots\cup\mathcal{Y}_{m}=\mathcal{Y}$$ and $$\mathcal{Y}_{i}\cap\mathcal{Y}_{j}=\emptyset \ \forall i\not=j$$