Goppa Codes. In this session, we will talk about another family of codes that have an efficient decoding algorithm: the Goppa codes. One limitation of the generalized Reed-Solomon codes is the fact that the length is bounded by the size of the field over which it is defined. This implies that these codes are useful when we use a large field size.
|Published (Last):||10 March 2013|
|PDF File Size:||6.70 Mb|
|ePub File Size:||6.43 Mb|
|Price:||Free* [*Free Regsitration Required]|
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. Patterson algorithm converts a syndrome to a vector of errors. 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. 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.
Binary Goppa code
Such codes were introduced by Valerii Denisovich Goppa. In particular cases, they can have interesting extremal properties. They should not be confused with binary Goppa codes that are used, for instance, in the McEliece cryptosystem. The vector space is a subspace of the function field of X. We usually denote a Goppa code by C D , G.