Article

Nombre de points rationnels de courbes algébriques sur des corps finis associées à des codes cycliques. (Number of rational points of algebraic curves over finite fields associated with cyclic codes)

January 1987
Comptes Rendus de l Académie des Sciences - Series I - Mathematics

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Article

Two Classes of Constacyclic Codes With Variable Parameters [( q m – 1)/ r , k , d ]

January 2023 · IEEE Transactions on Information Theory

Constacyclic codes over finite fields are a family of linear codes and contain cyclic codes as a subclass. Constacyclic codes are related to many areas of mathematics and outperform cyclic codes in several aspects. Hence, constacyclic codes are of theoretical importance. On the other hand, constacyclic codes are important in practice, as they have rich algebraic structures and may have efficient ... [Show full abstract] decoding algorithms. In this paper, two classes of constacyclic codes are constructed using a general construction of constacyclic codes with cyclic codes. The first class of constacyclic codes is motivated by the punctured Dilix cyclic codes and the second class is motivated by the punctured generalised Reed-Muller codes. The two classes of constacyclic codes contain optimal linear codes. The parameters of the two classes of constacyclic codes are analysed and some open problems are presented in this paper.

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Hasse–Weil bound for additive cyclic codes

January 2017 · Designs Codes and Cryptography

We obtain a bound on the minimum distance of additive cyclic codes via the number of rational points on certain algebraic curves over finite fields. This is an extension of the analogous bound in the case of classical cyclic codes. Our result is the only general bound on such codes aside from Bierbrauer’s BCH bound. We compare our bounds’ performance against the BCH bound for additive cyclic ... [Show full abstract] codes in a special case and provide examples where it yields better results.

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Generalized Hamming weights of trace codes

April 1994 · IEEE Transactions on Information Theory

Linear codes over F_p often admit a natural representation as trace codes of codes that are defined over an extension field F_pm. In the paper, the authors obtain estimates for the weights of subcodes of such trace codes. Their main result is a far-reaching generalization of the Carlitz-Uchiyama bound for the duals of binary BCH codes. In particular, they prove sharp bounds ... [Show full abstract] for the generalized Hamming weights of a large class of codes, including duals of BCH codes, classical Goppa codes, Melas codes, and arbitrary cyclic codes of length n=p^m-1. The main tool is the theory of algebraic functions over finite fields, in particular the Hasse-Weil bound for the number of places of degree one

Article

Packing superballs from codes and algebraic curves

January 2008 · Acta Mathematica Sinica

In the present paper, we make use of codes with good parameters and algebraic curves over finite fields with many rational points to construct dense packings of superballs. It turns out that our packing density is quite reasonable. In particular, we improve some values for the best-known lower bounds on packing density.

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Towers of Global Function Fields with Asymptotically Many Rational Places and an Improvement on the...

January 1998 · Mathematische Nachrichten

We construct infinite class field towers of global function fields with asymptotically many rational places. In this way, we improve on asymptotic bounds of Serre, Perret, Schoof, and Xing. The results can be interpreted equivalently as asymptotic bounds on the number of rational points of smooth algebraic curves over finite fields. As an application, we show an improvement on the ... [Show full abstract] Gilbert-Varshamov bound for linear codes over finite fields of a sufficiently large composite nonsquare order.

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New Linear Codes and Algebraic Function Fields Over Finite Fields

January 2008 · IEEE Transactions on Information Theory

In this correspondence, we present 129 new linear codes over F₈ and F₉ based on the construction by Xing and Niederreiter using algebraic function fields and places of small degrees. In addition, we construct some global function fields in which the number of rational places improves the lower bounds given by van der Geer and van der Vlugt.

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Elements of Prescribed Order, Prescribed Traces and Systems of Rational Functions Over Finite Fields

January 2005 · Designs Codes and Cryptography

Ferruh Özbudak

Let k ≥ 1 and

f{_1}, \ldots, f{_r} \in {\mathbb F}_{q^k}(x)

be a system of rational functions forming a strongly linearly independent set over a finite field

{\mathbb F}_q

. Let

\gamma_1, \ldots, \gamma_r \in {\mathbb F}_q

be arbitrarily prescribed elements. We prove that for all sufficiently large extensions

{\mathbb F}_{q^{km}}

, there is an element

\xi \in {\mathbb F}_{q^{km}}

... [Show full abstract] of prescribed order such that {\rm Tr}_{{\mathbb F}_{q^{km} }/{\mathbb F}_q}(f_i(\xi))=\gamma_i$ for $i=1, \ldots, r$, where ${\rm Tr}_{{\mathbb F}_{q^{km}/{\mathbb F}_q}} is the relative trace map from

{\mathbb F}_{q^{km}}

onto

{\mathbb F}_q

We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et~al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265–282) completely.

Conference Paper

On the Construction of Some Towers over Finite Fields

January 2003 · Lecture Notes in Computer Science

Jörg Wulftange

The explicit construction of towers with many rational places plays a key role for the construction of asymptotically good algebraic geometric codes. One way of explicitly constructing towers is given by defining them recursively via a single equation. In this paper we discuss conditions on the defining equation to give towers with many rational places and introduce a new family of such towers.

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Wiman’s and Edge’s sextics attaining Serre’s bound

March 2018 · European Journal of Mathematics

Motoko Qiu Kawakita

We find new explicit algebraic curves of genus 6 over the finite fields F673,F5393

\mathbb {F}_{67^3},\mathbb {F}_{5393}

and ... [Show full abstract] F9173

\mathbb {F}_{9173}

attaining Serre’s bound. They are Wiman’s and Edge’s sextics. Also we obtain a condition on p for Wiman’s sextic over the finite field Fp2

\mathbb {F}_{p^2}

to be maximal, and give new entries of genus 6 in the tables at manYPoints.org by computer search on Edge’s sextics.

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The set of pure gaps at several rational places in function fields

December 2024 · Designs Codes and Cryptography

In this work, we explore the use of maximal elements in generalized Weierstrass semigroups and their relationship with pure gaps, extending the results in Castellanos et al. [J Pure Appl Algebra 228(4):107513, 2024]. We provide a method to completely determine the set of pure gaps at several rational places in a function field F over a finite field, where the periods of certain places are the ... [Show full abstract] same, and determine its cardinality. As an example, we calculate the cardinality and provide a simple, explicit description of the set of pure gaps at several rational places distinct from the infinity place on Kummer extensions, offering a different characterization from that presented by Hu and Yang [Des Codes Cryptogr 86(1):211–230, 2018]. Furthermore, we present some applications in coding theory and AG codes with good parameters.

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Preprint

Weights in Codes and Genus 2 Curves

June 2003

We discuss a class of binary cyclic codes and their dual codes. The minimum distance is determined using algebraic geometry, and an application of Weil's theorem. We relate the weights appearing in the dual codes to the number of rational points on a family of genus 2 curves over a finite field.

Article

Multiple-Burst Error-Correcting Cyclic Product Codes

December 1987 · IEEE Transactions on Information Theory

W. van Overveld

Dimension 10 with minimal distance 9, and a code of length 21 and dimension 16 with minimal distance 3. The codes are constructed from algebraic geometry using the dictionary between coding theory and algebraic curves over finite fields established by Goppa. The curve used in the present work is the Klein quartic. This curve has the maximal number of rational points over GF(23) allowed by Serre's ... [Show full abstract] improvement of the Hasse-Weil bound, which, together with the low genus, accounts for the good code parameters. The Klein quartic has the Frobenius group G of order 21 acting as a group of automorphisms which accounts for the particular symmetries of the codes. In fact, the codes are given alternative descriptions as left ideals in the group-algebra GF(23)[G]. This description allows for easy decoding. For instance, in the case of the single error correcting code of length 21 and dimension 16 with minimal distance 3, decoding is obtained by multiplication with an idempotent in the group algebra.

Nombre de points rationnels de courbes algébriques sur des corps finis associées à des codes cycliques. (Number of rational points of algebraic curves over finite fields associated with cyclic codes)

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