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

Matthews, Gretchen L.

doi:10.1007/978-3-540-24633-6_2

Cited by 49 publications

(87 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…, m}. The next result shows that H (V, P m ) may be obtained from (V, P m ) using the operation lub (in the case where V is the canonical linear system see [22,Sect. 2] for a proof of a similar result and see [17] for a more detailed study of (V, P 2 ); cf.…”

Section: Definition 44 Ifmentioning

confidence: 92%

See 1 more Smart Citation

On Weierstrass semigroups and sets: a review with new results

Carvalho

Kato

2008

Geom Dedicata

View full text Add to dashboard Cite

In this work we present a survey of the main results in the theory of Weierstrass semigroups at several points, with special attention to the determination of bounds for the cardinality of its set of gaps. We also review results on applications to the theory of error correcting codes. We then recall a generalization of the concept of Weierstrass semigroup, which is the Weierstrass set associated to a linear system and several points. We finish by presenting new results on this Weierstrass set, including some on the cardinality of its set of gaps.

show abstract

Section: Definition 44 Ifmentioning

confidence: 92%

“…In [21] Matthews determined H (P 1 , P 2 ) for all pairs of distinct rational points on the Hermitian curve y q + y = x q+1 defined over F q 2 ; in [22] working with this same curve, she determined H (P m ) for any m ∈ {2, . .…”

Section: Weierstrass Semigroups At Several Pointsmentioning

confidence: 99%

On Weierstrass semigroups and sets: a review with new results

Carvalho

Kato

2008

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…To illustrate these techniques, we determine the Weierstrass semigroup of certain r-tuples of points on a quotient of the Hermitian curve for 2 ≤ r ≤ q + 1. This is a generalization of the main result of [15] where the Weierstrass semigroup of an r-tuple of collinear points on a Hermitian curve is described.…”

Section: Introductionmentioning

confidence: 91%

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

Matthews

2005

Des Codes Crypt

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…We note that Theorem 3.6 may be used to derive the Weierstrass gap set of any m-tuple consisting of distinct places of the form P ∞ and P , of a Hermitian function field where ∈ F q 2 is fixed. For another approach to determining this Weierstrass gap set, see [10,11]. It would also be possible to derive the set of pure gaps of m-tuples of the form (P ∞ , P , 2 , .…”

Section: Corollary 37mentioning

confidence: 99%

Riemann–Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences

Maharaj

Matthews

Pirsic

2005

Journal of Pure and Applied Algebra

Self Cite

View full text Add to dashboard Cite

This paper is concerned with two applications of bases of Riemann-Roch spaces. In the first application, we define the floor of a divisor and obtain improved bounds on the parameters of algebraic geometry codes. These bounds apply to a larger class of codes than that of Homma and Kim (J. Pure Appl. Algebra 162 (2001) 273). Then we determine explicit bases for large classes of Riemann-Roch spaces of the Hermitian function field. These bases give better estimates on the parameters of a large class of m-point Hermitian codes. In the second application, these bases are used for fast implementation of Xing and Niederreiter's method (Acta. Arith. 72 (1995) 281) for the construction of low-discrepancy sequences.

show abstract

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

Cited by 49 publications

References 9 publications

On Weierstrass semigroups and sets: a review with new results

On Weierstrass semigroups and sets: a review with new results

Weierstrass Semigroups and Codes from a Quotient of the Hermitian Curve

Riemann–Roch spaces of the Hermitian function field with applications to algebraic geometry codes and low-discrepancy sequences

Contact Info

Product

Resources

About