Long Quadratic Residue Codes
نویسنده
چکیده
A long standing problem has been to develop “good” binary linear block codes, C, to be used for error-correction. The length of the block is denoted n and the dimension of the code is denoted k. So in this notation C ⊆ GF (2) is a k-dimensional subspace. Another important parameter is the smallest weight of any non-zero codeword, d. This is related to error-correction because C can correct [d−1 2 ] errors. We will construct an interesting family of codes Ci which have the property that log2(|Ci|) n has a limit = 1 4 and d n has a limit ≥ 14 . We will also investigate how this family violates the “fake Goppa conjecture.” Moreover, we can show that the Goppa conjecture is incompatible with a conjecture on hyperelliptic curves over finite fields, which we call the “small content conjecture.”
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