Codes and Gap Sequences of Hermitian Curves

Korchmáros, Gábor; Nagy, Gábor P.; Timpanella, Marco

doi:10.1109/tit.2019.2950207

Cited by 12 publications

(7 citation statements)

References 17 publications

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“…{(0, 1), (0, 16), (1, 0), (2, 2), (2, 15), (3,6), (3,11), (5, 8), (5, 9), (6, 3), (6,14), (8, 5), (8,12), (9, 5), (9,12), (11,3), (11,14), (12,8), (12,9), (14,6), (14,11), (15,2), (15,15), (16, 0)}.…”

Section: A Small Examplementioning

confidence: 99%

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Goppa codes over Edwards curves

Filippone

2021

Preprint

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Section: A Small Examplementioning

confidence: 99%

“…Let K = GF(17) and let ❊ (K) be the Edwards curve of equation x 2 + y 2 = 1 − 8x 2 y 2 . There are 12 points on this curve: 15), (3,6), (3,11), (5,8), (5,9), (6, 3), (6,14), (8,5), (8,12), (9, 5), (9,12), (11,3), (11,14), (12,8), (12,9), (14,6), (14,11), (15,2), (15,15), (16, 0)}.…”

Section: A Small Examplementioning

confidence: 99%

Goppa codes over Edwards curves

Filippone

2021

Preprint

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“…Korchmáros and Nagy constructed P GU 3 -invariant q 6 -ary codes of length q 9 − q 3 [20], but the invariance is proved only for rates converging to zero. Eid et al construct q 4 -ary Suzuki-invariant codes of length q 4 + √ 2q 2.5 (q − 1) − q 2 [21] and dimension l(q 2 + 1) − q 0 (q − 1) + 1 for l ≤ q 2 − 1.…”

Section: Unitary Suzuki and Ree Codesmentioning

confidence: 99%

Capacity-achieving codes: a review on double transitivity

Ivanov,

Urbanke

2020

Preprint

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Recently it was proved that if a linear code is invariant under the action of a doubly transitive permutation group, it achieves the capacity of erasure channel. Therefore, it is of sufficient interest to classify all codes, invariant under such permutation groups. We take a step in this direction and give a review of all suitable groups and the known results on codes invariant under these groups. It turns out that there are capacity-achieving families of algebraic geometric codes.

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“…AG codes are proven to have good performances provided that X , G and D are carefully chosen in an appropriate way. In particular, AG codes with better parameters can arise from curves which have many q -rational points, especially from maximal curves which are curves defined over q with q square whose number of q -rational points X( q ) attains the Hasse-Weil upper bound, namely �X( q )� = q + 1 + 2 √ q , where is the genus of X ; for AG codes from maximal curves see for instance [6,13,17,18]. Regarding the choice of the two divisors D and G, the latter is typically taken to be a multiple mP of a single point P of degree one.…”

Section: Introductionmentioning

confidence: 99%

AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve

Lia

Timpanella

2021

AAECC

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In Beelen and Montanucci (Finite Fields Appl 52:10–29, 2018) and Giulietti and Korchmáros (Math Ann 343:229–245, 2009), Weierstrass semigroups at points of the Giulietti–Korchmáros curve $${\mathcal {X}}$$ X were investigated and the sets of minimal generators were determined for all points in $${\mathcal {X}}(\mathbb {F}_{q^2})$$ X ( F q 2 ) and $${\mathcal {X}}(\mathbb {F}_{q^6})\setminus {\mathcal {X}}( \mathbb {F}_{q^2})$$ X ( F q 6 ) \ X ( F q 2 ) . This paper completes their work by settling the remaining cases, that is, for points in $${\mathcal {X}}(\overline{\mathbb {F}}_{q}){\setminus }{\mathcal {X}}( \mathbb {F}_{q^6})$$ X ( F ¯ q ) \ X ( F q 6 ) . As an application to AG codes, we determine the dimensions and the lengths of duals of one-point codes from a point in $${\mathcal {X}}(\mathbb {F}_{q^7}){\setminus }{\mathcal {X}}( \mathbb {F}_{q})$$ X ( F q 7 ) \ X ( F q ) and we give a bound on the Feng–Rao minimum distance $$d_{ORD}$$ d ORD . For $$q=3$$ q = 3 we provide a table that also reports the exact values of $$d_{ORD}$$ d ORD . As a further application we construct quantum codes from $$\mathbb {F}_{q^7}$$ F q 7 -rational points of the GK-curve.

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Codes and Gap Sequences of Hermitian Curves

Cited by 12 publications

References 17 publications

Goppa codes over Edwards curves

Goppa codes over Edwards curves

Capacity-achieving codes: a review on double transitivity

AG codes from $${{\mathbb{F}}_{q^7}}$$-rational points of the GK maximal curve

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