Random projection algorithm pseudo code

I am trying to apply Random Projections method on a very sparse dataset. I found papers and tutorials about Johnson Lindenstrauss method, but every one of them is full of equations which makes no meaningful explanation to me. For example, this document on Johnson-Lindenstrauss

Unfortunately, from this document, I can get no idea about the implementation steps of the algorithm. It's a long shot but is there anyone who can tell me the plain English version or very simple pseudo code of the algorithm? Or where can I start to dig this equations? Any suggestions?

For example, what I understand from the algorithm by reading this paper concerning Johnson-Lindenstrauss is that:

1. Assume we have a AxB matrix where A is number of samples and B is the number of dimensions, e.g. 100x5000. And I want to reduce the dimension of it to 500, which will produce a 100x500 matrix.

As far as I understand: first, I need to construct a 100x500 matrix and fill the entries randomly with +1 and -1 (with a 50% probability).

Edit:
Okay, I think I started to get it. So we have a matrix A which is mxn. We want to reduce it to E which is mxk.

What we need to do is, to construct a matrix R which has nxk dimension, and fill it with 0, -1 or +1, with respect to 2/3, 1/6 and 1/6 probability.

After constructing this R, we'll simply do a matrix multiplication AxR to find our reduced matrix E. But we don't need to do a full m