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ag codes and ag quantum codes from the GGS curve

DESIGNS codeS AND CRYPTOGRAPHY 2018年第10期86卷 2315-2344页

作者： Bartoli, Daniele Montanucci, Maria Zini, Giovanni Univ Perugia Dipartimento Matemat & Informat Via Vanvitelli 1 I-06123 Perugia Italy Univ Basilicata Dipartimento Matemat Informat & Econ Campus Macchia RomanaViale Ateneo Lucano 10 Potenza Italy Univ Firenze Dept Matemat & Informat Florence Italy

In this paper, algebraic-geometric (ag) codes associated with the GGS maximal curve are investigated. The Weierstrass semigroup at all -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.

关键词： GGS curve ag code Quantum code Convolutional code code automorphisms

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Multi point ag codes on the GK maximal curve

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DESIGNS codeS AND CRYPTOGRAPHY 2018年第1期86卷 161-177页

作者： Bartoli, Daniele Montanucci, Maria Zini, Giovanni Univ Perugia Dipartimento Matemat & Informat Via Vanvitelli 1 I-06123 Perugia Italy Univ Basilicata Dipartimento Matemat Informat & Econ Campus Macchia RomanaViale Ateneo Lucano 10 I-85100 Potenza Italy Univ Florence Dipartimento Matemat & Informat Viale Morgagni 67-A I-50134 Florence Italy

In this paper we investigate multi-point Algebraic-Geometric codes associated to the GK maximal curve, starting from a divisor which is invariant under a large automorphism group of the curve. We construct families of... 详细信息

关键词： GK curve ag code code automorphisms

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Fast Encoding of ag codes Over Cab Curves

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IEEE TRANSACTIONS ON INFORMATION THEORY 2021年第3期67卷 1641-1655页

作者： Beelen, Peter Rosenkilde, Johan Solomatov, Grigory Tech Univ Denmark Dept Appl Math & Comp Sci DK-2800 Lyngby Denmark

We investigate algorithms for encoding of one-point algebraic geometry (ag) codes over certain plane curves called C-ab curves, as well as algorithms for inverting the encoding map, which we call "unencoding". Some C-ab curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity (O) over tilde (n(3/2)) resp. (O) over tilde (qn) for ag codes over any C-ab curve satisfying very mild assumptions, where n is the code length and q the base field size, and (O) over tilde ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity (O) over tilde (n) for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities (O) over tilde (n(5/4)) and (O) over tilde (n(3/2)) respectively.

关键词： Encoding ag code Hermitian code C-ab code norm-trace curve

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New Lower Bounds on the Generalized Hamming Weights of ag codes

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IEEE TRANSACTIONS ON INFORMATION THEORY 2014年第10期60卷 5930-5937页

作者： Bras-Amoros, Maria Lee, Kwankyu Vico-Oton, Albert Univ Rovira & Virgili Dept Comp Engn & Math E-43007 Tarragona Catalonia Spain Chosun Univ Dept Math & Educ Kwangju 501759 South Korea

A sharp upper bound for the maximum integer not belonging to an ideal of a numerical semigroup is given and the ideals attaining this bound are characterized. Then, the result is used, through the so-called Feng-Rao numbers, to bound the generalized Hamming weights of algebraic-geometry codes. This is further developed for Hermitian codes and the codes on one of the Garcia-Stichtenoth towers, as well as for some more general families.

关键词： Numerical semigroup ideal of a semigroup ag code isometry-dual sets of ag codes generalized Hamming weights order bound Feng-Rao number Hermitian codes Garcia-Stichtenoth towers

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Isometry-dual flags of ag codes

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DESIGNS codeS AND CRYPTOGRAPHY 2020年第8期88卷 1617-1638页

作者： Bras-Amoros, Maria Duursma, Iwan Hong, Euijin Univ Rovira & Virgili Tarragona Catalonia Spain Univ Illinois Urbana IL USA Univ Colorado Boulder CO 80309 USA

Consider a complete flag code = C-0 2g + 2 rational points is isometry-dual if and only if the last code C-n in the flag is defined with functions of pole order at most n + 2g - 1. Using a different approach, we exten... 详细信息

Consider a complete flag ag code = C-0 < C-1 < center dot center dot center dot < C-n = F-n of one-point ag codes of length n over the finite field F. The codes are defined by evaluating functions with poles at a given point Q in points P-1, ... , P-n distinct from Q. A flag has the isometry-dual property if the given flag and the corresponding dual flag are the same up to isometry. For several curves, including the projective line, Hermitian curves, Suzuki curves, Ree curves, and the Klein curve over the field of eight elements, the maximal flag, obtained by evaluation in all rational points different from the point Q, is self-dual. More generally, we ask whether a flag obtained by evaluation in a proper subset of rational points is isometry-dual. In Geil et al. (2011) it is shown, for a curve of genus g, that a flag of one-point ag codes defined with a subset of n > 2g + 2 rational points is isometry-dual if and only if the last code C-n in the flag is defined with functions of pole order at most n + 2g - 1. Using a different approach, we extend this characterization to all subsets of size n >= 2g + 2. Moreover we show that this is best possible by giving examples of isometry-dual flags with n = 2g + 1 such that Cn is generated by functions of pole order at most n + 2g - 2. We also prove a necessary condition, formulated in terms of maximum sparse ideals of the Weierstrass semigroup of Q, under which a flag of punctured one-point ag codes inherits the isometry-dual property from the original unpunctured flag.

关键词： ag code Punctured code Dual code

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Isometry-Dual Flags of Many-Point ag codes

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SIAM JOURNAL ON APPLIED ALGEBRA AND GEOMETRY 2023年第4期7卷 786-808页

作者： Bras-Amoros, Maria Castellanos, Alonso S. Quoos, Luciane Univ Rovira & Virgili Av Paisos Catalans 26 Tarragona 43007 Catalonia Spain Univ Fed Uberlandia Fac Matemat Bloco 1FAve Joao Naves Avila 2121 BR-38408100 B Santa Monica Brazil Univ Fed Rio de Janeiro Ctr Tecnol Cidade UnivAve Athos Silveira Ramos 149 BR-21941909 Rio De Janeiro Brazil

A flag of linear codes C0 ( C1 ( \cdot \cdot \cdot ( Cs is said to have the isometry-dual property if there exists a vector \bfx \in (Fqast \)n such that Ci = \bfx \cdot C\bot s - i, where Cibot \denotes the dual code of the code Ci. We extend our previous results in [M. Bras-Amoro'\s, A. S. Castellanos, and L. Quoos, IEEE Trans. Inform. Theory, 68 (2022), pp. 828--838] of flags of algebraic geometry two-point codes over a function field F to flags of (t + 1)-point codes C\scrL (D, a0P + \sum ti=1 \beta iQi) ( C\scrL (D, a1P + \sum ti=1 \beta iQi)) ( \cdot \cdot \cdot ( C\scrL (D,asP +\sum t i=1 \beta iQi) for any tuple of integers \beta 1, ... ,\beta t and for an increasing sequence of integers a0, . . . , as, just provided that n \geq 2g + 2, where g is the genus of F. We apply the obtained results to the broad class of Kummer extensions defined by affine equations of the form ym = f (x) for f(x), a separable polynomial of degree r, where gcd(r, m) = 1. In particular, we obtain necessary and sufficient conditions on m and \beta i's such that the flag has the isometry-dual property.

关键词： ag code function field dual code flag of codes isometry-dual property

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Rational points on cubic surfaces and ag codes from the Norm-Trace curve

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ANNALI DI MATEMATICA PURA ED APPLICATA 2023年第1期202卷 185-208页

作者： Bonini, Matteo Sala, Massimiliano Vicino, Lara Aalborg Univ Dept Math Sci Aalborg Denmark Univ Trento Dept Math Trento Italy Tech Univ Denmark Dept Appl Math & Comp Sci Lyngby Denmark

In this paper, we derive general bounds for the number of rational points on a cubic surface defined over F-q, which constitute an extension of a result due to Weil. Exploiting these bounds, we are able to give a complete characterization of the intersections between the Norm-Trace curve over F-q3 and the curves of the form y = ax(3) + bx(2) + cx + d, generalizing a previous result by Bonini and Sala and providing more detailed information about the weight spectrum of one-point ag codes arising from such curve.

关键词： Norm-Trace curve ag code Weight spectrum Cubic surfaces

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

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JOURNAL OF GEOMETRY 2018年第1期109卷 23-23页

作者： Montanucci, Maria Timpanella, Marco Zini, Giovanni Univ Basilicata Potenza Italy Univ Milano Bicocca Milan Italy

We investigate several types of linear codes constructed from two families of maximal curves over finite fields recently constructed by Skabelund as cyclic covers of the Suzuki and Ree curves. Plane models for such cu... 详细信息

关键词： Suzuki curve Ree curve ag code Quantum code Convolutional code code automorphisms

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

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IEEE TRANSACTIONS ON INFORMATION THEORY 2022年第2期68卷 828-838页

作者： Bras-Amoros, Maria Castellanos, Alonso Sepulveda Quoos, Luciane Univ Rovira & Virgili Dept Comp Engn & Math Tarragona 43002 Catalonia Spain Univ Fed Uberlandia Fac Matemat BR-38408100 Uberlandia MG Brazil Univ Fed Rio de Janeiro Inst Matemat BR-21941901 Rio De Janeiro Brazil

A flag of codes C-0 not subset of C-1 not subset of center dot center dot center dot not subset of C-s not subset of F-q(n) is said to satisfy the isometry-dual property if there exists x is an element of (F-q*)(n) such that the code C-i is x-isometric to the dual code C-s-i(perpendicular to) for all i = 0, ..., s. For P and Q rational places in a function field F, we investigate the existence of isometry-dual flags of codes in the families of two-point algebraic geometry codes C-L(D, a(0)P + bQ) not subset of C-L(D, a(1)P + bQ) not subset of center dot center dot center dot not subset of C-L(D, a(s)P + bQ), where the divisor D is the sum of pairwise different rational places of F and P, Q are not in supp(D). We characterize those sequences in terms of b for general function fields. We then apply the result to the broad class of Kummer extensions F defined by affine equations of the form y(m) = f(x), for f(x) a separable polynomial of degree r, where gcd(r, m) = 1. For P the rational place at infinity and Q the rational place associated to one of the roots of f(x), and for D an Aut(F/F-q)-invariant sum of rational places of F, such that P, Q is not an element of supp D, it is shown that the flag of two-point algebraic geometry codes has the isometry-dual property if and only if m divides 2b + 1. At the end we illustrate our results by applying them to two-point codes over several well know function fields.

关键词： ag code function field dual code flag of codes isometry-dual property

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

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DESIGNS codeS AND CRYPTOGRAPHY 2020年第8期88卷 1595-1616页

作者： Castellanos, Alonso Sepulveda Bras-Amoros, Maria Univ Fed Uberlandia Uberlandia MG Brazil Univ Rovira & Virgili Tarragona Catalonia Spain

We determine the Weierstrass semigroup H(P infinity,P1, horizontal ellipsis ,Pm) at several rational points on the maximal curves which cannot be covered by the Hermitian curve introduced in Tafazolian et al. (J Pure Appl Algebra 220(3):1122-1132, 2016). Furthermore, we present some conditions to find pure gaps. We use this semigroup to obtain ag codes with better relative parameters than comparable one-point ag codes arising from these curves.

关键词： Weierstrass semigroup Maximal curve ag code

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