On Weierstrass semigroups and sets: a review with new results

Matthews and Michel \cite{Michel} investigated the minimum distances of certain algebraic-geometry codes arising from a higher degree place $P$. In terms of the Weierstrass gap sequence at $P$, they proved a bound that gives an improvement on the designed minimum distance. In this paper, we consider those of such codes which are constructed from the Hermitian function field $\mathbb F_{q^2}(\HC)$. We determine the Weierstrass gap sequence $G(P)$ where $P$ is a degree $3$ place of $\mathbb F_{q^2}(\HC)$, and compute the Matthews and Michel bound with the corresponding improvement. We show more improvements using a different approach based on geometry. We also compare our results with the true values of the minimum distances of Hermitian $1$-point codes, as well as with estimates due to Xing and Chen \cite{XC}

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“…From the above discussion we have the following result. where δ is the designed minimum distance of the code given in (5).…”

Section: Background and Preliminary Resultsmentioning

confidence: 99%

“…Remark 5.3. From (8), a lower bound for the minimum distance of (7) is δ + 3(q + 1 − m 0 ) with δ designed minimum distance given in (5).…”

Section: Improvements On the Matthews-michel Boundmentioning

confidence: 99%

Hermitian codes from higher degree places

Korchmáros

Nagy

2013

Journal of Pure and Applied Algebra

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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%

ORDINARY SPECIAL WEIERSTRASS n-SEMIGROUPS

Ballico¹

2015

Int. J. of Pure and Appl. Math.

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“…. , m. The set of pure gaps of X at Q will be denoted by G 0 (Q); for further information concerning these objects, we refer the reader to the survey [4].…”

Section: Introductionmentioning

confidence: 99%

On Weierstrass Gaps at Several Points

Tenório

Tizziotti

2018

Bull Braz Math Soc, New Series

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We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado [7], we present a description of these elements which generalizes the approach of Homma and Kim [11] given for pairs. Through this description, we study the gaps and pure gaps at several points on a certain family of curves with separated variables.Date: March 26, 2018.

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On Weierstrass semigroups and sets: a review with new results

Cited by 21 publications

References 24 publications

Hermitian codes from higher degree places

Hermitian codes from higher degree places

ORDINARY SPECIAL WEIERSTRASS n-SEMIGROUPS

On Weierstrass Gaps at Several Points

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