Hello,
I've been playing around with linear error-correcting codes with the intention of programming a demonstration of McEliece's Cryptosystem, which utilises Goppa codes.
I've been following the format outlined in the SAGE tutorial. I can't post the link for some reason, but it can be found by searching "sagemath Goppa codes". I have no issue running the following code to create a Goppa code over field GF(2^4) and its corresponding decoder.
F = GF(2^4)
R.<x> = F[]
g = x^2+x+1
L = [a for a in F.list if g(a) != 0]
C = codes.GoppaCode(g,L)
D = C.decoder()
However, the cryposystem I'm trying to program requires parameters higher than this, specifically over the field of GF(2^10). The code that I've used to create this Goppa code and its decoder are therefore as follows.
F = GF(2^10)
R.<x> = F[]
g = x^50+x^3+1
L = [a for a in F.list if g(a) != 0]
C = codes.GoppaCode(g,L)
D = C.decoder()
This creates a [1024, 524] Goppa code over GF(2) and there's no issue in generating the code C, or even using it to encode a message vector (of length 524).
Unfortunately, I seem to encounter a memory issue when I run the final line to create the decoder D. My initial response was to try it on a machine with larger memory (64GB), but I had no luck there. I assume it's because the code that I'm trying to create is pretty huge, but it would be really helpful if there's somehow a way around the problem to allow me implement this Goppa code and generate its decoder.
Any help would be greatly appreciated and I'm happy to provide more details if needed.
Thank you very much!!
