I explained all the linear algebra involved in the Peterson-Gorenstein-Zierler decoder and offer a working implementation in https://www.nayuki.io/page/reed-sol

I explained all the linear algebra involved in the Peterson-Gorenstein-Zierler decoder and offer a working implementation in https://www.nayuki.io/page/reed-solomon-error-correcting-code-decoder https://www.nayuki.io/page/reed-solomon-error-correcting-cod... ; https://news.ycombinator.com/item?id=13565939 https://news.ycombinator.com/item?id=13565939 It runs in O(n^3) time, and unfortunately is not the state of the art. What actual Reed-Solomon implementations use are the Berlekamp-Massey and Forney algorithms, which run in O(n^2) time.