Codes from Goppa codes

Abstract

On a Goppa code whose structure polynomial has coefficients in the symbol field, the Frobenius acts. Its fixed codewords form a subcode. Deleting the naturally occurred redundance, we obtain a new code. It is proved that these new codes approach the Gilbert-Varshamov bound. It is also proved that these codes can be decoded within $O(n^2(\logn)^a)$ operations in the symbol field, which is usually much small than the location field, where $n$ is the codeword length, and $a$ a constant determined by the polynomial factorization algorithm.