Codes From Algebraic Geometry
Codes obtained from algebraic curves have attracted much attention from mathematicians and engineers alike since the remarkable work of Tsfasman et al.  who showed that the longstAnding Gilbert-Varshamov lower bound can be exceeded for alphabet sizes larger than 49. The Gilbert-Varshamov bound, established in 1952, is a lower bound on the information rate of good codes. This lower bound was not improved until 1982 with the discovery of good algebraic geometric codes. These codes are obtained from modular curves , but consideration of these curves is beyond the scope of this book. van Lint and Springer  later derived the same results as Tsfasman et al., but by using less complicated concepts from algebraic geometry. Recall that a linear code with parameters [n, k, d] q is a linear subspace of F q n of dimension k and minimum distance d.
KeywordsElliptic Curve Elliptic Curf Linear Code Algebraic Curf Parity Check Matrix
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