erasure_code_test can not support specific case

[liujiangang01]

if k=9,m=18, and column in (0, 1, 2, 5, 7, 9, 10, 14, 16) corrupt, isa cannot decode
when execute example, report as follows:
erasure_code_test: 127x8192 BAD MATRIX
Fail to gf_gen_decode_matrix

[liujiangang01]

测试代码参数调整：


测试代码参数调整成k=9,m=18，并且模拟0, 1, 2, 5, 7, 9, 10, 14, 16列损坏，运行结果（报错）如下：

[gbtucker]

I don't think this is a bug but you are pushing the limits of a Vandermonde matrix. As described in the header for gf_gen_rs_matrix for k = 9 you are not guaranteed an invertable matrix unless m <= 13. You might consider using a different encoding matrix like a Cauchy-based one the example uses. You can test this will work with:

$ ./ec_simple_example -k 9 -p 9 -e 0 -e 1 -e 2 -e 5 -e 7 -e 9 -e 10 -e 14 -e 16
ec_simple_example:
 encode (m,k,p)=(18,9,9) len=8192
 recover 9 fragments
 check recovery of block { 0 1 2 5 7 9 10 14 16 } done all: Pass

[liujiangang01]

Thanks，yesterday I use Cauchy-based matrix， it does work

[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