Goppa code

Binary Goppa code is defined by a polynomial g(x) of degree t over a finite field without multiple zeros, and a sequence L of n distinct elements from , that aren't roots of the polynomial:



Codewords belong to the kernel of syndrome function, forming a subspace of :



Code defined by a tuple (g,L) has minimum distance 2t+1, thus it can correct errors in a word of size (n-mt) using codewords of size n. It also possesses a convenient parity-check matrix H in form: