Clarify gf_gen_rs_matrix documentation

[asomers]

In the PDF documentation for gf_gen_rs_matrix, there is a list of inequalities which will guarantee that every submatrix of the result will be invertible. However, the inequalities are rendered in such a way that they all kind of run together. Maybe there are some missing commas? Or maybe the inequalities come in pairs? It's really unclear. Could you please clarify it?

[gbtucker]

There is a test function included called gen_rs_matrix_limits that is intended to help explore the bounties where sub-matrices are invertible for gf_gen_rs_matrix up to some max. It should make it more clear as it prints for each k the highest m found up to max search. Note that the encoding generator functions gf_gen_rs_matrix() and gf_gen_cauchy1_matrix() are only two examples of how to pick an encoder and the function ec_encode_data() will work for any encoding matrix.

[gbtucker]

Sorry, I realize what you mean now about the pdf version. Doxygen was collapsing newlines and running them together. We pushed a fix in latest.

[jasonkresch]

There is a way to generate a systematic encoding matrix that is guaranteed to be invertible in any case. It works as follows:

Step 1. Generate a K by K+M Vandermonde matrix
Step 2. Generate a K by K Vandermonde matrix
Step 3. Invert the K by K Vandermonde matrix generated in step 2
Step 4. Multiply the matrix generated step 1 by the inverse Vandermonde matrix created in step 3

The matrix resulting from step 4 can serve as an encoding matrix. The first K x K rows will correspond to the identity matrix. The remaining bottom M rows can serve the purposes of encoding redundancy.

For reference, this is how the matrix is generated in other Reed-Solomon libraries, such as zfec:
https://github.com/tahoe-lafs/zfec/blob/master/zfec/fec.c#L420