On Goppa Codes and Weierstrass Gaps at Several Points

Abstract

We generalize results of Homma and Kim [J.

Algebraic Geometry Codes: General Theory

“…With the Klein curve over F 8 , and for G = 2∆, we found a code of type [24,4,19]. In this case, D ∼ 8∆ and the code with G = 6∆ is of type [24,16,7]. This is the best known three-error-correcting code of length 24 over F 8 .…”

“…The top row in the table below gives the weight distribution for a code of type [24,16,7] over F 8 constructed with the Klein curve. The method outlined here produces the weight distributions for all 2744 = 14 3 codes of type [24,16] on the Klein curve.…”

Weierstrass Semigroup and Pure Gaps at Several Points on the GK Curve

“…For this curve, taking m = 1, by Theorem 3.4, we have that (2,11), (3,3), (4, 13), (5, 5), (7, 7), (10, 10), (11,2), (13,4), (19, 1)} .…”

“…Kim in [11]. In [2], C. Carvalho and F. Torres used the concept of pure gaps to obtain codes whose minimum distance have bounds better than the Goppa bound. Using the concept of discrepancy we have the following result to obtain pure gaps.…”

The Heaviest Weierstrass N-Semigroups on Non-Hyperelliptic Curves?

“…. , a n ) ∈ N n such that there is a rational function on X with a 1 P 1 + · · · + a n P n as its divisor of poles ( [1], [2], [3]) The set H(P 1 , . .…”