Good and asymptotically good quantum codes derived from algebraic geometry

Guardia, Giuliano G. La; Pereira, Francisco Revson Fernandes

doi:10.1007/s11128-017-1618-7

Cited by 22 publications

(26 citation statements)

References 27 publications

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“…In this section we construct quantum codes from families of one-point AG codes onS q , using the so-called CSS construction which allows to construct quantum codes from classical linear codes; see [32,Lemma 2.5]. Let q be a prime power and H = (C q ) ⊗n = C q ⊗· · ·⊗C q be a q n -dimensional Hilbert space.…”

Section: Quantum Codes Froms Qmentioning

confidence: 99%

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AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

2018

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Section: Quantum Codes Froms Qmentioning

confidence: 99%

“…As an application of Lemma 5.1 to AG codes, La Guardia and Pereira proposed in [32] the following general t-point construction.…”

Section: Quantum Codes Froms Qmentioning

confidence: 99%

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

2018

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“…In this section we use families of one-point AG codes from the GGS curve to construct quantum codes. The main ingredient is the so called CSS contruction which enables to construct quantum codes from classical linear codes; see [29,Lemma 2.5].…”

Section: Quantum Codes From One-point Ag Codes On the Ggs Curvesmentioning

confidence: 99%

“…If δ Q = 0, then the code is said to be quantum MDS. For a detailed introduction on quantum codes see [29] and the references therein. Lemma 6.1.…”

Section: Quantum Codes From One-point Ag Codes On the Ggs Curvesmentioning

confidence: 99%

AG codes and AG quantum codes from the GGS curve

Bartoli

Montanucci

Zini

2017

Des. Codes Cryptogr.

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In this paper, algebraic-geometric (AG) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all F q 2 -rational points of the curve is determined; the Feng-Rao designed minimum distance is computed for infinite families of such codes, as well as the automorphism group. As a result, some linear codes with better relative parameters with respect to one-point Hermitian codes are discovered. Classes of quantum and convolutional codes are provided relying on the constructed AG codes. support of G. Maximal curves over F q attain the Hasse-Weil upper bound for the number of F q -rational points with respect to their genus and for this reason they have been used in a number of works. Examples of such curves are the Hermitian curve, the GK curve [12], the GGS curve [10], the Suzuki curve [7], the Klein quartic when √ q ≡ 6 (mod 7) [33], together with their quotient curves. Maximal curves often have large automorphism groups which in many cases can be inherited by the code: this can bring good performances in encoding [25] and decoding [17]. Good bounds on the parameters of one-point codes, that is AG codes arising from divisors G of type nP for a point P of the curve, have been obtained by investigating the Weierstrass semigroup at P . These results have been later generalized to codes and semigroups at two or more points; see e.g. [4,5,20,21,27,30,31].AG codes from the Hermitian curve have been widely investigated; see [8,[22][23][24]37,39,40] and the references therein. Other constructions based on the Suzuki curve and the curve with equation y q + y = x q r +1 can be found in [32] and [36]. More recently, AG Codes from the GK curve have been constructed in [1,3,9].In the present work we investigate one-point AG codes from the F q 2n -maximal GGS curve, n ≥ 5 odd. The GGS curve has more short orbits under its automorphism group than other maximal curves, see [15], and hence more possible structures for the Weierstrass semigroups at one point. On the one hand this makes the investigation more complicated; on the other hand it gives more chances of finding one-point AG codes with good parameters. One achievement of this work is the determination of the Weierstrass semigroup at any F q 2 -rational point.We show that the one-point codes at the infinite point P ∞ inherit a large automorphism group from the GGS curve; for many of such codes, the full automorphism group is obtained. Moreover, for q = 2, we compute explicitly the Feng-Rao designed minimum distance, which improves the Goppa designed minimum distance. As an application, we provide families of codes with q = 2 whose relative Singleton defect goes to zero as n goes to infinity. We were not able to produce analogous results for an F q 2 -rational affine point P 0 , because of the more complicated structure of the Weierstrass semigroup. In a comparison between onepoint codes from P ∞ and one-point codes from P 0 , it turns out that the best codes come sometimes from P ∞ , other times from P 0 ; we give evidence of this fact with...

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“…[17, Lemma 2.5] Let C 1 and C 2 be two linear codes with parameters[N, k i , d i ] q , i = 1, 2, and assume that C 1 ⊂ C 2 . Then there exists an [[N, k 2 − k 1 , D]] q -code with D = min{wt(c)|c ∈ (C 2 \ C 1 ) ∪ (C ⊥ 1 \ C ⊥ 2 )}, where wt(c) is the weight of c.…”

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confidence: 99%

AG codes from the second generalization of the GK maximal curve

Montanucci

Lavorante

2020

Discrete Mathematics

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The second generalized GK maximal curves GK 2,n [1] are maximal curves over finite fields with q 2n elements, where q is a prime power and n ≥ 3 an odd integer. In this paper we determine the structure of the Weierstrass semigroup H(P ) where P is an arbitrary F q 2 -rational point of GK 2,n . We show that these points are Weierstrass points and the Frobenius dimension of GK 2,n is computed. A new proof of the fact that the first and the second generalized GK curves are not isomorphic for any n ≥ 5 is obtained. AG codes and AG quantum codes from the curve GK 2,n are constructed; in some cases, they have better parameters with respect to those already known.

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Good and asymptotically good quantum codes derived from algebraic geometry

Cited by 22 publications

References 27 publications

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

AG codes and AG quantum codes from cyclic extensions of the Suzuki and Ree curves

AG codes and AG quantum codes from the GGS curve

AG codes from the second generalization of the GK maximal curve

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