If you're curious, you might also be interested in Cauchy-Reed Solomon coding. This converts Galois field operations into XORs by treating elements of GF(2^n) as bit matrices. The advantage then is that instead of doing Galois field operations, you can just xor things for much better performance. The canonical paper is https://web.eecs.utk.edu/~jplank/plank/papers/CS-05-569.pdf.

https://www.usenix.org/system/files/fast19-zhou.pdf is a more modern paper that goes into some related problems of trying to reduce the number of XOR operations needed to encode data.