The theta divisor of a jacobian variety and the decoding of geometric Goppa codes

AlgebraicGeometry Codes

“…The Klein curve goes back to F. Klein [54] and has been studied thoroughly, also over finite fields in connection with codes. See [17,19,25,43,46,62].…”

On the decoding of algebraic-geometric codes

“…This curve has been studied most often in papers on AG codes and their decoding [16,18,25,38,43,63,81,98]. In these papers however one considers codes of the form…”

“…Furthermore s = O(n) and the complexity of the algorithm is O(n 4 ) for n → ∞. [43,81]. It looks like a difficult (possibly hopeless) problem in general, which is, moreover, obsolete from the decoding point of view, given the solutions of the decoding problem [17,22,23] which we will discuss in Sections 7 and 8.…”

“…The smallest value for s was studied by Carbonne, Henocq, Rotillon and Thiong-Ly [11,42,43]. This algorithm is not effective, that is to say the existence of these s divisors is guaranteed by a counting argument using the zeta function of the curve, but the actual construction of those was only achieved for hyperelliptic curves of genus at most 4 by Le Brigand [58,59] and the Klein curve by Henocq and Rotillon [43,81]. It looks like a difficult (possibly hopeless) problem in general, which is, moreover, obsolete from the decoding point of view, given the solutions of the decoding problem which we will discuss below.…”