Binary Goppa code

In mathematics and computer science, the binary Goppa code is an error-correcting code that belongs to the class of general Goppa codes originally described by Valerii Denisovich Goppa, but the binary structure gives it several mathematical advantages over non-binary variants, also providing a better fit for common usage in computers and telecommunication. Binary Goppa codes have interesting properties suitable for cryptography in McEliece-like cryptosystems and similar setups.

A binary Goppa code is defined by a polynomial ${\displaystyle g(x)}$ of degree ${\displaystyle t}$ over a finite field ${\displaystyle GF(2^{m})}$ without multiple zeros, and a sequence ${\displaystyle L}$ of ${\displaystyle n}$ distinct elements from ${\displaystyle GF(2^{m})}$ that aren't roots of the polynomial:

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