Add RS decoder

[jonas-lj]

Implement a RS decoder with Gao's algorithm. This relies on implementations of polynomial division and extended GCD for polynomials, so the polynomial module has been extended accordingly.

[arnab-roy]

Jonas, I ran some additional tests on the decoder. It's producing correct results and seems to be pretty efficient around n = O(100). However, it's running ~300-400s on my machine for n=1000. Is that acceptable or we want to optimize more? I can push the new tests to your branch if that helps.

[jonas-lj]

Jonas, I ran some additional tests on the decoder. It's producing correct results and seems to be pretty efficient around n = O(100). However, it's running ~300-400s on my machine for n=1000. Is that acceptable or we want to optimize more? I can push the new tests to your branch if that helps.

That's a good point. The implementation is rather naive, so there are a few low hanging optimisations, we could try. If it breaks that much for n=1000, I suspect just using iterators could be a worthwhile optimisation. I'll work on it and get back to you.

[jonas-lj]

Jonas, I ran some additional tests on the decoder. It's producing correct results and seems to be pretty efficient around n = O(100). However, it's running ~300-400s on my machine for n=1000. Is that acceptable or we want to optimize more? I can push the new tests to your branch if that helps.

That's a good point. The implementation is rather naive, so there are a few low hanging optimisations, we could try. If it breaks that much for n=1000, I suspect just using iterators could be a worthwhile optimisation. I'll work on it and get back to you.

@arnab-roy I found the bottleneck and got the decoding down to 120 ms for n=1024 which should be fine.

[arnab-roy]

@jonas-lj Amazing! I tested your new code and now the same tests are 1-2s in my machine (likely more than the numbers you got due to crude measurements and machine specs). Ran some more edge cases as well, and all working :)

[jonas-lj]

@jonas-lj Amazing! I tested your new code and now the same tests are 1-2s in my machine (likely more than the numbers you got due to crude measurements and machine specs). Ran some more edge cases as well, and all working :)

Yea, the trick to get a 99% improvement in performance is just to start from something really bad #til