Geometric Goppa Codes Research Articles

On Geometric Goppa Codes from Elementary Abelian p-Extensions of $${{\mathbb{F}}}_{{p}^{s}}(x)$$

Let p be a prime number and s > 0 an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian p-extension of \({{\mathbb{F}}}_{{p}^{s}}(x)\). We determine their dimension and exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for self-duality and quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, and locally recoverable codes on the function field.

A Note on the Rank Bounded Distance and Its Algorithms for Cyclic Codes

The minimum distance for linear codes is one of the important parameters. The shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. It is necessary to construct the maximum value of the independent set for the calculation of the shift bound. However, its computational complexity is very large, because the construction of the independent set is not unique. The authors proposed an algorithm for calculation of the independent set using the discrete Fourier transform in 2010. In this paper we give simple modification and new recurrent algorithms to improve the original algorithm.

On the duals of geometric Goppa codes from norm-trace curves

In this paper we study the dual codes of a wide family of evaluation codes on norm-trace curves. We explicitly find out their minimum distance and give a lower bound for the number of their minimum-weight codewords. A general geometric approach is performed and applied to study in particular the dual codes of one-point and two-point codes arising from norm-trace curves through Goppaʼs construction, providing in many cases their minimum distance and some bounds on the number of their minimum-weight codewords. The results are obtained by showing that the supports of the minimum-weight codewords of the studied codes obey some precise geometric laws as zero-dimensional subschemes of the projective plane. Finally, the dimension of some classical two-point Goppa codes on norm-trace curves is explicitely computed.

A Note on the Shift Bound for Cyclic Codes by the DFT

For cyclic codes some well-known lower bounds and some decoding methods up to the half of the bounds are suggested. Particularly, the shift bound is a good lower bound of the minimum distance for cyclic codes, Reed-Muller codes and geometric Goppa codes. In this paper we consider cyclic codes defined by their defining set, and new simple derivation of the shift bound using the discrete Fourier transform with unknown elements and the Blahut theorem is shown. Moreover two examples of binary cyclic codes are given.

Near Orders and Codes

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> Høholdt, van Lint, and Pellikaan used order functions to construct codes by means of Linear Algebra and Semigroup Theory only. However, Geometric Goppa codes that can be represented by this method are mainly those based on just one point. In this correspondence, we introduce the concept of near order function with the aim of generalizing this approach in such a way that a wider family of Geometric Goppa codes can be studied on a more elementary setting. </para>

The parity check codes through geometric Goppa codes.

This work treats a way to obtain the parity check codes through geometric Goppa codes. Two cases have been considered depending on the taken field characteristic as alphabet and on the code length. It has been used the technique of linear code restriction. More precisely, it has been showed that the parity check codes are obtained as the restriction of a rational geometric Goppa code.

Algebraic geometric codes with applications

The theory of linear error-correcting codes from algebraic geometric curves (algebraic geometric (AG) codes or geometric Goppa codes) has been well-developed since the work of Goppa and Tsfasman, Vladut, and Zink in 1981–1982. In this paper we introduce to readers some recent progress in algebraic geometric codes and their applications in quantum error-correcting codes, secure multi-party computation and the construction of good binary codes.

A generalized floor bound for the minimum distance of geometric Goppa codes

We prove a new bound for the minimum distance of geometric Goppa codes that generalizes two previous improved bounds. We include examples of the bound applied to one- and two-point codes over certain Suzuki and Hermitian curves.

Toward an Explicit Construction of Nonlinear Codes Exceeding the Tsfasman–VlĂduŢ–Zink Bound

We consider asymptotically good nonlinear codes recently introduced by Xing (2003). The original definition of these codes relies on a nonconstructive averaging argument. In this paper, it is first shown that in some cases, the codes can be constructed without using any averaging arguments. We then introduce an alternative construction of the codes, based on the union of a geometric Goppa code and its cosets. In some cases, the problem of explicitly describing the codes reduces to the problem of explicitly describing certain n elements of the relevant function field, where n is the code length. Moreover, the number of finite-field operations required to construct these n elements after the construction of the generator matrix of the geometric Goppa code is of the order of n/sup 3/.

On codes from norm–trace curves

The main results of this paper are derived by using only simple Gröbner basis techniques. We present a new construction of evaluation codes from Miura–Kamiya curves C ab . We estimate the minimum distance of the codes and estimate the minimum distance of a class of related one-point geometric Goppa codes. With respect to these estimates the new codes perform at least as well as the related geometric Goppa codes. In particular we consider codes from norm–trace curves. We show that our estimates give actually the true minimum distance of these codes. The new codes from norm–trace curves perform rather well. In many cases much better than the corresponding geometric Goppa codes. It turns out that an alternative description of the new codes from norm–trace curves can be made by using Høholdt et al.'s in: V.S. Pless, W.C. Huffman (Eds.), Handbook of Coding Theory, Vol. 1, Elsevier, Amsterdam, 1998, pp. 871–961 (Chapter 10) construction of improved dual codes.

Sudan-decoding generalized geometric Goppa codes

Generalized geometric Goppa codes are vector spaces of n-tuples with entries from different extension fields of a ground field. They are derived from evaluating functions similar to conventional geometric Goppa codes, but allowing evaluation in places of arbitrary degree. A decoding scheme for these codes based on Sudan's improved algorithm is presented and its error-correcting capacity is analyzed. For the implementation of the algorithm it is necessary that the so-called increasing zero bases of certain spaces of functions are available. A method to obtain such bases is developed.

GENERALIZED GEOMETRIC GOPPA CODES

ABSTRACT Conventional geometric Goppa codes are defined in terms of functions of an algebraic function field associated with a divisor evaluated in places of degree 1. The generalization that will be treated here allows evaluations in places of arbitrary degree. With the appropriate inner product, the dual of the code can be defined and described in terms of Weil differentials similarly to conventional geometric Goppa codes. A decoding algorithm is derived.

Lower Bounds on the State Complexity of Geometric Goppa Codes

We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if \deg G\leq \frac{n-2}{2} or \deg G\geq \frac{n-2}{2}+2g then the state complexity of C_{\mathcal{L}}(D,G) is equal to the Wolf bound. For \deg G\in[\frac{n-1}{2},\frac{n-3}{2}+2g], we use Clifford's theorem to give a simple lower bound on the state complexity of C_{\mathcal{L}}(D,G). We then derive two further lower bounds on the state space dimensions of C_{\mathcal{L}}(D,G) in terms of the gonality sequence of F/\Bbb{F}_q. (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of C_{\mathcal{L}}(D,G) and one follows in a straightforward way from first principless often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.

On the existence of order functions

The notions of well-behaving sequences and order functions is fundamental in the elementary treatment of geometric Goppa codes. The existence of order functions is proved with the theory of Grobner bases.

Weierstrass Pairs and Minimum Distance of Goppa Codes

We prove that elements of the Weierstrass gap set of a pair of points may be used to define a geometric Goppa code which has minimum distance greater than the usual lower bound. We determine the Weierstrass gap set of a pair of any two Weierstrass points on a Hermitian curve and use this to increase the lower bound on the minimum distance of particular codes defined using a linear combination of the two points.

The true dimension of certain alternant codes

By applying a result from geometric Goppa codes, due to H.Schtenoth, the true dimension of certain alternant codes is calculated. The results lead in many cases to an improvement of the usual lower bound for the dimension.

Construction of Improved Geometric Goppa Codes from Klein Curves and Klein-like Curves

Linear error-correcting codes, especially Reed-Solomon codes, find applications in communication and computer memory systems, to enhance their reliability and data integrity. In this paper, we present Improved Geometric Goppa (IGG) codes, a new class of error-correcting codes, based on the principles of algebraic-geometry. We also give a reasonably low complexity procedure for the construction of these IGG codes from Klein curves and Klein-like curves, in plane and high-dimensional spaces. These codes have good code parameters like minimum distance rate and information rate, and have the potential to replace the conventional Reed-Solomon codes in most practical applications. Based on the approach discussed in this paper, it might be possible to construct a class of codes whose performance exceeds the Gilbert-Varshamov bound.

Computing Weierstrass Semigroups and the Feng–Rao Distance from Singular Plane Models

We present an algorithm to compute the Weierstrass semigroup at a point P together with functions for each value in the semigroup, provided P is the only branch at infinity of a singular plane model for the curve. As a byproduct, the method also provides us with a basis for the spaces L(mP) and the computation of the Feng–Rao distance for the corresponding array of geometric Goppa codes. A general computation of the Feng–Rao distance is also obtained. Everything can be applied to the decoding problem by using the majority scheme of Feng and Rao.

On generalized hyperbolic cascaded Reed–Solomon codes

In this paper, a generalized version of the Hyperbolic Cascaded Reed–Solomon (HCRS) code proposed by K. Saints and C. Heegard is considered. The lower bound of the minimum distance (the design distance) is investigated. This is a linear code that can be expressed as (n, k, d min = d ) on GF(q). Here, n, k, dmin, and d are the code length, dimension, minimum distance, and design distance, respectively. The proposed code has more degrees of freedom in selection of the code length than the HCRS code. Codes with various code length can be constructed easily. However, in the m-dimensional space on GF(q), the selectable code length is less than qm. When the proposed code with a code length less than qm is compared with the shortened code with the same code length obtained by shortening the m-dimensional HCRS code with a code length of qm [or (q−1)m], the design distances are identical, whereas the proposed code has an equal or greater dimension compared with the HCRS code. Also, from the proposed code, it is relatively easy to construct a code of the same code length as the algebraic geometric code (geometric Goppa code) and the improved geometric Goppa code proposed by Feng and Rao. Further, when the performance (parameters) is compared at a high coding rate (k/n), the proposed code sometimes has equal or better performance. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(11): 18–27, 1999

On generalized hyperbolic cascaded Reed-Solomon codes

In this paper, a generalized version of the Hyperbolic Cascaded Reed–Solomon (HCRS) code proposed by K. Saints and C. Heegard is considered. The lower bound of the minimum distance (the design distance) is investigated. This is a linear code that can be expressed as (n, k, d min = d ) on GF(q). Here, n, k, dmin, and d are the code length, dimension, minimum distance, and design distance, respectively. The proposed code has more degrees of freedom in selection of the code length than the HCRS code. Codes with various code length can be constructed easily. However, in the m-dimensional space on GF(q), the selectable code length is less than qm. When the proposed code with a code length less than qm is compared with the shortened code with the same code length obtained by shortening the m-dimensional HCRS code with a code length of qm [or (q−1)m], the design distances are identical, whereas the proposed code has an equal or greater dimension compared with the HCRS code. Also, from the proposed code, it is relatively easy to construct a code of the same code length as the algebraic geometric code (geometric Goppa code) and the improved geometric Goppa code proposed by Feng and Rao. Further, when the performance (parameters) is compared at a high coding rate (k/n), the proposed code sometimes has equal or better performance. © 1999 Scripta Technica, Electron Comm Jpn Pt 3, 82(11): 18–27, 1999