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.
An irreducible binary Goppa code is defined by a polynomial
with no repeated roots, and a sequence
Codewords belong to the kernel of the syndrome function, forming a subspace of
: The code defined by a tuple
, thus it can encode messages of length at least
It possesses a convenient parity-check matrix
in form Note that this form of the parity-check matrix, being composed of a Vandermonde matrix
, shares the form with check matrices of alternant codes, thus alternant decoders can be used on this form.
Such decoders usually provide only limited error-correcting capability (in most cases
For practical purposes, parity-check matrix of a binary Goppa code is usually converted to a more computer-friendly binary form by a trace construction, that converts the
binary matrix by writing polynomial coefficients of
Decoding of binary Goppa codes is traditionally done by Patterson algorithm, which gives good error-correcting capability (it corrects all
design errors), and is also fairly simple to implement.
Patterson algorithm converts a syndrome to a vector of errors.
The syndrome of a binary word
is expected to take a form of Alternative form of a parity-check matrix based on formula for
can be used to produce such syndrome with a simple matrix multiplication.
, but that is the case when the input word is a codeword, so no error correction is necessary.
using the extended euclidean algorithm, so that
Finally, the error locator polynomial is computed as
Note that in binary case, locating the errors is sufficient to correct them, as there's only one other value possible.
In non-binary cases a separate error correction polynomial has to be computed as well.
If the original codeword was decodable and the
was the binary error vector, then Factoring or evaluating all roots of
Binary Goppa codes viewed as a special case of Goppa codes have the interesting property that they correct full
errors in ternary and all other cases.
Asymptotically, this error correcting capability meets the famous Gilbert–Varshamov bound.
Because of the high error correction capacity compared to code rate and form of parity-check matrix (which is usually hardly distinguishable from a random binary matrix of full rank), the binary Goppa codes are used in several post-quantum cryptosystems, notably McEliece cryptosystem and Niederreiter cryptosystem.