Computing Weierstrass Semigroups and the Feng–Rao Distance from Singular Plane Models

Abstract

We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at in"nity of a singular plane model for the curve. As a byproduct, the method also provides us with a basis for the spaces L(mP) and the computation of the Feng}Rao distance for the corresponding array of geometric Goppa codes. A general computation of the Feng}Rao distance is also obtained. Everything can be applied to the decoding problem by using the majority scheme of Feng and Rao. Academic Press

On the generalized Feng-Rao numbers of numerical semigroups generated by intervals

“…There are some well-known facts about the functions ν and δ F R for an arbitrary semigroup S (see [11], [12] or [2] for further details). An important one is that δ F R (m) ≥ m + 1 − 2g for all m ∈ S with m ≥ c, and that equality holds if moreover m ≥ 2c − 1 (see also Proposition 9).…”

“…Observe that x − S contains all the integers not greater than x − c and that the number of integers smaller than x not belonging to x − S is precisely the genus of S. As the number of non-negative integers not greater than x is x + 1, one gets immediately the well known fact (see [11], [12] or [2]):…”

“…This problem has been broadly studied in the literature for different types of semigroups (see [2], [3] or [12]). In numerical terms, the above mentioned improvement of the Goppa distance in coding theory means the following: For a semigroup S with genus g and m ∈ S the Feng-Rao distance satisfies δ F R (m + 1) ≥ m + 2 − 2g if m > 2g − 2, and equality holds for m >> 0.…”

Evaluation codes defined by finite families of plane valuations at infinity

“…δ-sequences in N >0 were introduced by Abhyankar and Moh to study the semigroup at infinity of plane curves with only one place at infinity. Some interesting references concerning these curves are [5,3,4,37,33,41,13] and, concerning coding theory, Weierstrass semigroups and the least order bound of the Goppa codes, attached to curves whose plane models are curves as mentioned, were computed by Campillo and Farran in [7]. …”

“…Indeed, e 0 = (21, 9, 21, 9), m 0 = (35,15,35,15) and [(35, 15, 35, 15), (21,9,21,9)] is in the class 1; 1, 3 . Moreover, d 2 = (7, 3, 7, 3) because (35, 15, 35, 15) = 2(14, 6, 14, 6) + (7,3,7,3) and (14, 6, 14, 6) = 2(7, 3, 7, 3), therefore n 1 = 5 and so e 1 = (7, 3, 7, 3) and m 1 = (21, 8, 15, 11). Finally, we complete our explanation after checking that (m 1 , e 1 ) is in the class represented by 2, 1, ∞ .…”