Fast Encoding of AG Codes over $C_{ab}$ Curves

Abstract

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 Õpn 3{2 q resp. Õpqnq 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 Õ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, notably the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity Õpnq for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding algorithm has complexity Õpn 5{4 q while unencoding has Õpn 3{2 q.