Non-special divisors on Kummer extensions and applications to AG code hulls

For an AG code C = C(D, G) on a curve X /Fq , the structure of the hull (the intersection of a code with its dual) plays an important role in code equivalence, entanglement-assisted quantum codes, and efficient decoding. In general the hull need not be an AG code, but it becomes one when a certain divisor A associated to G and its dual divisor H is non-special. This reduces the problem to determining when effective divisors of the form A = \sum_{P in X} \max{v_P(G), v_P(H)} P are non special. In this talk I give explicit criteria for non-speciality of several classes of effective divisors of small degree on Kummer extensions. The characterization is obtained using the description of the Weierstrass semigroup at multiple points of these curves.

These results yield new families where the hull of an AG code can be written again as an AG code, and illustrate how the geometry of Kummer extensions controls the Riemann–Roch behaviour of low-degree divisors.