Weierstrass Multiple Loci ofn-Pointed Algebraic Curves

Ballico, Edoardo; Kim, S.J

doi:10.1006/jabr.1997.7166

Cited by 24 publications

(24 citation statements)

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“…This study was carried on by S. J. Kim [7] and M. Homma [5]. The Weierstrass gap set of an m-tuple of points where m ≥ 2 has been examined by E. Ballico and Kim [2], and more recently, by C. Carvalho and F. Torres [3]. Weierstrass gap sets play an interesting role in the construction and analysis of algebraic geometry codes (see [4], [9], [6], [3]).…”

Section: Introductionmentioning

confidence: 99%

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The Weierstrass Semigroup of an m-tuple of Collinear Points on a Hermitian Curve

Matthews

2004

Lecture Notes in Computer Science

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Abstract. We examine the structure of the Weierstrass semigroup of an m-tuple of points on a smooth, projective, absolutely irreducible curve X over a finite field IF. A criteria is given for determining a minimal subset of semigroup elements which generate such a semigroup where 2 ≤ m ≤| IF |. For all 2 ≤ m ≤ q + 1, we determine the Weierstrass semigroup of any m-tuple of collinear IF q 2 -rational points on a Hermitian curve y q + y = x q+1 .

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Section: Introductionmentioning

confidence: 99%

“…By Theorem 10, Γ (1,10,37), (1,19,28), (1,28,19), (1,37,10), (1, 46, 1), (2, 2, 38), (2,11,29), (2,20,20), (2, 29, 11), (2, 38, 2), (3,3,30), (3,12,21), (3,21,12), (3,30,3), (4,4,22), (4, 13, 13), (4,22,4), (5,5,14), (5,14,5), (6,6,6), (10, 1, 37), (10,10,28), (10,19,19), (10,28,10), (10, …”

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confidence: 97%

The Weierstrass Semigroup of an m-tuple of Collinear Points on a Hermitian Curve

Matthews

2004

Lecture Notes in Computer Science

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“…In particular, if G is a divisor supported by r points, then one can use the Weierstrass semigroup of the r-tuple of these points to estimate the parameters of the associated r-point code. While the Weierstrass semigroup of an r-tuple of points is a generalization of the classically studied Weierstrass semigroup of a point, very little is known about this set if r ≥ 2 (see [2,8,11]). The only families of curves over a finite field for which the Weierstrass semigroup of even a pair of points has been determined are the families of hyperelliptic and plane quartic curves [11], Hermitian [14], and Suzuki curves [13].…”

Section: Introductionmentioning

confidence: 99%

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

Matthews

2005

Des Codes Crypt

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We consider the quotient of the Hermitian curve defined by the equation y q + y = x m over F q 2 where m > 2 is a divisor of q + 1. For 2 ≤ r ≤ q + 1, we determine the Weierstrass semigroup of any r-tuple of F q 2 -rational points (P ∞ , P 0b 2 , . . . , P 0br ) on this curve. Using these semigroups, we construct algebraic geometry codes with minimum distance exceeding the designed distance. In addition, we prove that there are r-point codes, that is codes of the form C (D, α 1 P ∞ , α 2 P 0b 2 , + · · · + α r P 0br ) where r ≥ 2, with better parameters than any comparable one-point code on the same curve. Some of these codes have better parameters than comparable one-point Hermitian codes over the same field. All of our results apply to the Hermitian curve itself which is obtained by taking m = q + 1 in the above equation.

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“…. , 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]). The set H(P 1 , .…”

Section: Ordinary Weierstrass N-semigroupsmentioning

confidence: 99%