Generalized Hermitian codes
TL;DR: This work investigates one-point algebraic geometry codes defined from curves related to the Hermitian curve and obtains codes attaining new records on the parameters.
Abstract: We investigate one-point algebraic geometry codes defined from curves related to the Hermitian curve. We obtain codes attaining new records on the parameters.
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TL;DR: In this article, the authors investigated two families of maximal curves over finite fields constructed by Skabelund as cyclic covers of the Suzuki and Ree curves, and showed that S ˜ q is not Galois covered by the Hermitian curve maximal over F q 4, and R ˜ Q is not GCL covered by Hermitians maximal over f q 6.
22 citations
TL;DR: In this article, a relation between minimal value set polynomials defined over F q and certain q -Frobenius nonclassical curves is established, which leads to a characterization of the curves of type g ( y ) = f ( x ), whose irreducible components are q-Frosius non-classical.
12 citations
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TL;DR: In this article, a relation between minimal value set polynomials and Frobenius nonclassical curves is established, and the connection leads to a characterization of the curves of type $g(y)=f(x) whose irreducible components are $q$-Frobeniusnonclassical.
Abstract: We establish a relation between minimal value set polynomials defined over $\mathbb{F}_q$ and certain $q$-Frobenius nonclassical curves. The connection leads to a characterization of the curves of type $g(y)=f(x)$, whose irreducible components are $q$-Frobenius nonclassical. An immediate consequence will be the realization of rich sources of new $q$-Frobenius nonclassical curves.
9 citations
Cites background from "Generalized Hermitian codes"
...Some authors have used additional arithmetic properties of this curve to construct algebraic geometric codes with good parameters (see [4], [20], [21])....
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...Additional arithmetic properties of the curve GS ([4],[20], [21]) make it suitable for construction of algebraic geometric codes with good parameters....
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TL;DR: The main result is to find a basis of the Riemann-Roch space of a series of divisors, which can be used to construct multi-point codes explicitly, and an explicit formula enables one to calculate the parameters of these codes.
Abstract: We investigate multi-point algebraic geometric codes defined from curves related to the generalized Hermitian curve introduced by Bassa et al. Our main result is to find a basis of the Riemann–Roch space of a series of divisors, which can be used to construct multi-point codes explicitly. These codes turn out to have nice properties similar to those of Hermitian codes, for example, they are easy to describe, to encode and decode. It is shown that the duals are also such codes and an explicit formula is given. In particular, this formula enables one to calculate the parameters of these codes. Finally, we apply our results to obtain linear codes attaining new records on the parameters. A new record-giving [234, 141,≥ 59]-code over $ \mathbb {F}_{27} $ is presented as one of the examples.
9 citations
Cites background from "Generalized Hermitian codes"
... 20 25 y−5 3 21 26 x 1y−5 2 24 27 y−6 2 25 28 x 1y 6 1 Comparing Table II with the reference [22], we find the follow ing codes over F8 with the best known parameters: [28,1,28], [28,2,24], [28,3,24], [28,8,16], [28,12,12], [28,17,8], [28,25,3], [28,26,2], [28,27,2], [28,28,1]. Table II enables one to construct the codes explicitly. For instance, a [28,8,16]-code is constructed by the basisx2y, xy, x2, x, 1...
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...ed Hermitian curves proposed by Garcia and Stichtenoth; and calculated some parameters of these codes [15]. Some generalizations of these codes were studied by C. Munuera, A. Sepúlveda, and F. Torres [16]. In this paper we investigate multi-point codes from the other generalized Hermitian curves. Let q be a prime power, Fqn0 be the finite field of order qn0, with n0 >2, and j0,k0 be two relatively pr...
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TL;DR: In this paper, complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve are investigated.
Abstract: We investigate complete arcs of degree greater than two, in projective planes over finite fields, arising from the set of rational points of a generalization of the Hermitian curve. The degree of the arcs is closely related to the number of rational points of a class of Artin-Schreier curves which is calculated by using exponential sums via Coulter's approach. We also single out some examples of maximal curves.
7 citations
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TL;DR: MAGMA as mentioned in this paper is a new system for computational algebra, and the MAGMA language can be used to construct constructors for structures, maps, and sets, as well as sets themselves.
7,310 citations
Book•
25 Jun 1993
TL;DR: This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded and contains numerous exercises that help the reader to understand the basic material.
Abstract: The theory of algebraic function fields has its origins in number theory, complex analysis (compact Riemann surfaces), and algebraic geometry. Since about 1980, function fields have found surprising applications in other branches of mathematics such as coding theory, cryptography, sphere packings and others. The main objective of this book is to provide a purely algebraic, self-contained and in-depth exposition of the theory of function fields. This new edition, published in the series Graduate Texts in Mathematics, has been considerably expanded. Moreover, the present edition contains numerous exercises. Some of them are fairly easy and help the reader to understand the basic material. Other exercises are more advanced and cover additional material which could not be included in the text. This volume is mainly addressed to graduate students in mathematics and theoretical computer science, cryptography, coding theory and electrical engineering.
2,041 citations
TL;DR: Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced.
Abstract: The theory of error-correcting codes derived from curves in an algebraic geometry was initiated by the work of Goppa as generalizations of Bose-Chaudhuri-Hocquenghem (BCH), Reed-Solomon (RS), and Goppa codes. The development of the theory has received intense consideration since that time and the purpose of the paper is to review this work. Elements of the theory of algebraic curves, at a level sufficient to understand the code constructions and decoding algorithms, are introduced. Code constructions from particular classes of curves, including the Klein quartic, elliptic, and hyperelliptic curves, and Hermitian curves, are presented. Decoding algorithms for these classes of codes, and others, are considered. The construction of classes of asymptotically good codes using modular curves is also discussed.
304 citations
TL;DR: The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound and is explained in terms of linear algebra and the theory of semigroups only.
Abstract: The concept of an error-correcting array gives a new bound on the minimum distance of linear codes and a decoding algorithm which decodes up to half this bound. This gives a unified point of view which explains several improvements on the minimum distance of algebraic-geometric codes. Moreover, it is explained in terms of linear algebra and the theory of semigroups only.
199 citations
01 Jan 1992
150 citations