Chapter 4: Algebraic Geometry

In the last chapter, we discussed the Reed–Solomon codes, which are used to evaluate a polynomial of degree at most k − 1 at n(≥ k) points in the ring of polynomials F_q[x], where F_q is a field with q elements. We see that it is equivalent to evaluate L(D) on $P_{F_q}^1$ (see Example 1 of Section 5.1) with D = nP_∞ (see Section 4.3). Similarly, classical Goppa code can be considered as a code over $P_{F_q}^1$. We may extend the concept of Reed– Solomon codes and classical Goppa codes to codes over any projective smooth curve (instead of lines only). The Riemann–Roch theorem induces a richer algebraic structure, and the corresponding codes will be more useful…