Strange vectors and matrix notations

Question

I am reading a paper named "LDPC Codes for Compressed Sensing Alexandros G. Dimakis, Member, IEEE, Roxana Smarandache, Member, IEEE, and Pascal O. Vontobel, Member, IEEE". In that paper I came across strange vector and matrices notations. For example, $H=(h_{j,i})_{j,i}$ is some matrix, $\mathbf{a}=(a_{(i,m)})_{(i,m)}\in C^{Mn}$ is a vector. I am attaching an image of my paper. Can somebody help me with these symbols?Vector notation, Matrix Notation.

Answer 1

For the matrix, it is the standard notation where $h_{j,i}$ is the entry in the $j$-th row and $i$-th column, hence $(h_{j,i})_{j,i}$ is the collection of the $h_{j,i}$ indexed by the pairs $(j,i)$ which is the matrix you would expect, i.e. $$\begin{pmatrix}h_{1,1} & \dots & h_{1,I}\\\vdots &&\vdots \\ h_{J,1} & \dots & h_{J,I} \end{pmatrix}$$ if $i$ ranges from 1 to $I$ and $j$ from 1 to $J$.

For the vector, you have a finite set of indices $[n]\times[M]$ composed of pairs $(i,m)$. If you were to order those pairs, you would have a one-to-one correspondence with the set $\{1, \dots, nM\}$ and you could rewrite your vector as $\textbf{a} = (a_k)_{1\leq k\leq nM} = (a_1, \dots, a_{nM})$, which is probably what you are used to see. The fact here is that this requires choosing an ordering of $[n]\times[M]$ as this set is not canonically ordered so instead, you just index the entries by the pair $(i,m)$ which plays the role of $k$ in the ordered case except that you cannot write it $(a_1, \dots, a_{nM})$ as there is no prescribed order.