Fast Encoding of AG Codes over C_abCurves

We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called C{ab} curves, as well as algorithms for inverting the encoding map, which we call 'unencoding'. Some C{ab} curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity tilde { mathcal {O}}(text {n}{3/2}) resp. tilde { mathcal {O}}({ittext { qn}}) for AG codes over any C{ab} curve satisfying very mild assumptions, where n is the code length and q the base field size, and tilde { mathcal {O}} ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity tilde { mathcal {O}}(text {n}) for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities tilde { mathcal {O}}(text {n}{5/4}) and tilde { mathcal {O}}(text {n}{3/2}) respectively.