Weierstrass semigroup at $$m+1$$ rational points in maximal curves which cannot be covered by the Hermitian curve

Castellanos, Alonso S.; Bras-Amorós, Maria

doi:10.1007/s10623-020-00757-4

Cited by 4 publications

(4 citation statements)

References 30 publications

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“…The fact that (x − α) = m(P α − P ∞ ) shows that π ≤ m. Suppose that 1 ≤ π ≤ m − 1 and let z be a rational function such that (z) = π(P α − P ∞ ). By [10,Proposition 4.3], the smallest non-zero element in the Weierstrass semigroup H(P α ) at P α is m − ⌊m/r⌋. Therefore, m − ⌊m/r⌋ ≤ π ≤ m − 1.…”

Section: Characterizing the Divisors Giving The Isometrydual Propertymentioning

confidence: 99%

“…In this case the two flags of codes are different. However, they are related as shown in Equation (9). Computations show that in this case the vector v in that equation is…”

Section: Appendixmentioning

confidence: 99%

“…A function field over F q 2 is called a maximal function field if its number of rational places attains the Hasse-Weil upper bound q + 1 + 2gq. Constructions of AG codes over maximal function fields have been a proficuous object of research in the last decades, see [8,9,19,23] and references in there.…”

Section: Introductionmentioning

confidence: 99%

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The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

Bras-Amorós¹,

Castellanos

Quoos

2022

IEEE Trans. Inform. Theory

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q of linear codes is said to be self-orthogonal if the duals of the codes in the flag satisfy C ⊥ i = C s−i , and it is said to satisfy the isometry-dual property with respect to an isometry vector x if C ⊥ i = xC s−i for i = 1, . . . , s. We characterize complete (i.e. s = n) flags with the isometry-dual property by means of the existence of a word with non-zero coordinates in a certain linear subspace of F n q . For flags of algebraic geometry (AG) codes we prove a so-called translation property of isometry-dual flags and give a construction of complete self-orthogonal flags, providing examples of self-orthogonal flags over some maximal function fields. At the end we characterize the divisors giving the isometry-dual property and the related isometry vectors showing that for each function field there is only a finite number of isometry vectors and that they are related by cyclic repetitions.

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Section: Characterizing the Divisors Giving The Isometrydual Propertymentioning

confidence: 99%

“…In this case the two flags of codes are different. However, they are related as shown in Equation (9). Computations show that in this case the vector v in that equation is…”

Section: Appendixmentioning

confidence: 99%

See 1 more Smart Citation

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

Bras-Amorós¹,

Castellanos

Quoos

2022

IEEE Trans. Inform. Theory

Self Cite

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“…, P n ) if and only if L(D α ) = L(D α − P 1 − • • • − P n ). The Weierstrass semigroups have been determined for various collections of points on particular families of curves of interest in coding theory [5,4,10,11,26].…”

Section: Introductionmentioning

confidence: 99%

Triples of rational points on the Hermitian curve and their Weierstrass semigroups

Matthews¹,

Skabelund²,

Wills³

2020

Preprint

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In this paper, we study configurations of three rational points on the Hermitian curve over F q 2 and classify them according to their Weierstrass semigroups. For q > 3, we show that the number of distinct semigroups of this form is equal to the number of positive divisors of q + 1 and give an explicit description of the Weierstrass semigroup for each triple of points studied. To do so, we make use of two-point discrepancies and derive a criterion which applies to arbitrary curves over a finite field.

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Generalized Weierstrass semigroups at several points on certain maximal curves which cannot be covered by the Hermitian curve

Montanucci

Tizziotti

2022

Des. Codes Cryptogr.

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Weierstrass semigroup at $$m+1$$ rational points in maximal curves which cannot be covered by the Hermitian curve

Cited by 4 publications

References 30 publications

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

The Isometry-Dual Property in Flags of Two-Point Algebraic Geometry Codes

Triples of rational points on the Hermitian curve and their Weierstrass semigroups

Generalized Weierstrass semigroups at several points on certain maximal curves which cannot be covered by the Hermitian curve

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