Goppa Codes

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Binary Goppa code is defined by a polynomial g(x) of degree t over a finite field 3 45.pngwithout multiple zeros, and a sequence L of n distinct elements from 3 45.png, that aren't roots of the polynomial:

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Codewords belong to the kernel of syndrome function, forming a subspace of 3 47.png:

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Code defined by a tuple (g, L) has minimum distance 2t+1, thus it can correct 3 49.pngerrors in a word of size n-mt, using codewords of size n. It also possesses a convenient parity-check matrix H in form:

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