R. Pellikaan

• Dr.
• Universitair Docent at Eindhoven University of Technology

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can b...

This paper builds a novel bridge between algebraic coding theory and mathematical knot theory, with applications in both directions. We give methods to construct error-correcting codes starting from the colorings of a knot, describing through a series of results how the properties of the knot translate into code parameters. We show that knots can b...

The extended coset leader weight enumerator of the generalized Reed–Solomon [q+1,q-3,5]q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[q+1,q-3,5]_q$$\end{document} co...

Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional...

The extended coset leader weight enumerator of the generalized Reed-Solomon $[q + 1, q - 3, 5]_q$ code is computed. The computation is considered as a question in finite geometry. For this we need the classification of the points, lines and planes in the projective three space under projectivities that leave the twisted cubic invariant. A line in t...

Quantum error correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error correcting codes to achieve higher rates than the one obtained via traditional stabilizer...

We introduce the Poincaré polynomial of a linear q-ary code and its relation to the corresponding weight enumerator. The question of whether the Poincaré polynomial is a complete invariant is answered affirmatively for q = 2, 3 and negatively for q ≥ 4. Finally we determine this polynomial for MDS codes and, by means of a recursive formula, for bin...

This well-balanced text touches on theoretical and applied aspects of protecting digital data. The reader is provided with the basic theory and is then shown deeper fascinating detail, including the current state of the art. Readers will soon become familiar with methods of protecting digital data while it is transmitted, as well as while the data...

The hull $H(C)$ of a linear code $C$ is defined by $H(C)=C \cap C^\perp$. A linear code with a complementary dual (LCD) is a linear code with $H(C)=\{0\}$. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show...

Error-correcting pairs were introduced independently by Pellikaan and K\"otter as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise products of vectors, and are used for many well-known families of codes. In this paper, we define new types of vector products, extending the coordinatewise product, some...

We give polynomial time attacks on the McEliece public key cryptosystem-based either on algebraic geometry (AG) codes or on small co-dimensional subcodes of AG codes. These attacks consist in the blind reconstruction either of an error correcting pair (ECP), or an error correcting array (ECA) from the single data of an arbitrary generator matrix of...

The problem of identifying whether the family of cyclic codes is asymptotically good or not is a long-standing open problem in the field of coding theory. It is known in the literature that some families of cyclic codes such as BCH codes and Reed-Solomon codes are asymptotically bad, however in general the answer to this question is not known. A re...

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid.

This paper defines the q-analogue of a matroid and establishes several properties like duality, restriction and contraction. We discuss possible ways to define a q-matroid, and why they are (not) cryptomorphic. Also, we explain the motivation for studying q-matroids by showing that a rank metric code gives a q-matroid.

Error-correcting pairs were introduced independently by Pellikaan and K\"otter as a general method of decoding linear codes with respect to the Hamming metric using coordinatewise products of vectors, and are used for many well-known families of codes. In this paper, we define new types of vector products, extending the coordinatewise product, some...

Error-correcting pairs were introduced in 1988 by R. Pellikaan, and were found independently by R. K\"otter (1992), as a general algebraic method of decoding linear codes. These pairs exist for several classes of codes. However little or no study has been made for characterizing those codes. This article is an attempt to fill the vacuum left by the...

This paper investigates the generalized rank weights, with a definition implied by the study of the generalized rank weight enumerator. We study rank metric codes over $L$, where $L$ is a finite Galois extension of a field $K$. This is a generalization of the case where $K = \mathbb{F}_q$ and $L = \mathbb{F}_{q^m}$ of Gabidulin codes to arbitrary c...

This paper investigates the rank weight enumerator of a code over L, where L is a finite extension of a field K. This is a generalization of the case where K = Fq and L = Fqm of Gabidulin codes to arbitrary characteristic. We use the notion of counting polynomials, to define the (extended) rank weight enumerator, since in this generality the set of...

We give a polynomial time attack on the McEliece public key cryptosystem based on subcodes of algebraic geometry (AG) codes. The proposed attack reposes on the distinguishability of such codes from random codes using the Schur product. Wieschebrink treated the genus zero case a few years ago but his approach cannot be extent straightforwardly to ot...

We give a polynomial time attack on the McEliece public key cryptosystem based on algebraic geometry codes. Roughly speaking, this attacks runs in $O(n^4)$ operations in $\mathbb F_q$, where $n$ denotes the code length. Compared to previous attacks, allows to recover a decoding algorithm for the public key even for codes from high genus curves.

This paper addresses the question of retrieving the triple ${(\mathcal X,\mathcal P, E)}$ from the algebraic geometry code ${\mathcal C = \mathcal C_L(\mathcal X, \mathcal P, E)}$ , where ${\mathcal X}$ is an algebraic curve over the finite field ${\mathbb F_q, \,\mathcal P}$ is an n-tuple of ${\mathbb F_q}$ -rational points on ${\mathc...

This paper addresses the question how often the square code of an arbitrary l-dimensional subcode of the code GRSk (a, b) is exactly the code GRS2k-1(a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first used and stated by Wieschebrink where h...

Code-based cryptography is an interesting alternative to classic number-theoretic public key cryptosystem since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems such as algebraic geometry codes. In Márquez-Corbella et al. (2012) – for so called very strong algebraic g...

This paper considers the truncation of matroids and geometric lattices. It is shown that the truncated matroid of a representable matroid is again representable. Truncation formulas are given for the coboundary and Möbius polynomial of a geometric lattice and the spectrum polynomial of a matroid, generalizing the truncation formula of the rank gene...

Code-based cryptography is an interesting alternative to classic number-theory PKC since it is conjectured to be secure against quantum computer attacks. Many families of codes have been proposed for these cryptosystems, one of the main requirements is having high performance t-bounded decoding algorithms which in the case of having high an error-c...

This paper addresses the question of how often the square code of an arbitrary l-dimensional subcode of the code GRS k (a, b) is exactly the code GRS 2k−1 (a, b * b). To answer this question we first introduce the notion of gaps of a code which allows us to characterize such subcodes easily. This property was first stated and used in [10] where Wie...

We consider the weight distribution of the binary cyclic code of length 2^n-1 with two zeros @a^a,@a^b. Our proof gives information in terms of the zeta function of an associated variety. We carry out an explicit determination ...

We discuss decoding techniques and finding the minimum distance of linear codes with the use of Gröbner bases. First, we give a historical overview of decoding cyclic codes via solving systems of polynomial equations over finite fields. In particular, we mention papers of Cooper, Reed, Chen, Helleseth, Truong, Augot, Mora, Sala, and others. Some st...

The problem of bounded distance decoding of arbitrary linear codes using Gröbner bases is addressed. A new method is proposed, which is based on reducing an initial decoding problem to solving a certain system of polynomial equations over a finite field. The peculiarity of this system is that, when we want to decode up to half the minimum distance,...

In this short note we show how one can decode linear error-correcting codes up to half the minimum distance via solving a system of polynomial equations over a finite field. We also explicitly present the reduced Gröbner basis for the system considered.

This paper gives a survey on extended and generalized weight enumerators of a linear code and the Tutte polynomial of the matroid of the code [16]. Furthermore ongoing research is reported on the coset leader and list weight enumerator and its extensions using the derived code and its arrangement of hyperplanes.

We propose two simple and e-cient deterministic extractors for an or- dinary elliptic curve E, deflned overF2N, where N = 2' and ' is a positive integer. Our extractors, for a given point P on E, output respectively the flrst or the second F2'-coe-cient of the abscissa of the point P. We also propose two deterministic extractors for the main subgro...

We propose a simple and efficient deterministic extractor for the (hyper)elliptic curve \(\mathcal{C}\), defined over \(\mathbb{F}_{q^2}\), where q is some power of an odd prime. Our extractor, for a given point P on \(\mathcal{C}\), outputs the first \(\mathbb{F}_{q}\)-coefficient of the abscissa of the point P. We show that if a point P is chosen...

The problem of decoding up to error correcting capacity of arbi-trary linear codes with the use of Gröbner bases is addressed. A new method is proposed, which is based on reducing an initial decoding problem to solving some system of polynomial equations over a finite field. The peculiarity of this system is that, when we want to decode up to half...

Without Abstract

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the min- imum distance of a cyclic code. We use the AB-method to obtain a dierent bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class o...

The q-ary Reed-Muller (RM) codes RM_q(u,m) of length n=q^m are a generalization of Reed-Solomon (RS) codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the univariate case, randomized list-decoding algorithms for RM codes were given in an...

We present a variant of the Diffie-Hellman scheme in which the number of bits exchanged is one third of what is used in the classical Diffie-Hellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a extension of this variant further reducing the size of sent infor...

Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in and the corresponding weighted total-degree monomial orderings lift naturally from one domain Rj−1 in the tower to the next, Rj, the integral closure of Rj−1[xj]/〈φ(xj)〉. The qth power algorithm is reworked in this more general setting to pro...

The notion of an order domain is generalized. The behaviour of an order domain by taking a subalgebra, the extension of scalars, and the tensor product is studied. The relation of an order domain with valuation theory, Gröbner algebras, and graded structures is given. The theory of Gröbner bases for order domains is developed and used to show that...

Curves and surfaces of type I are generalized to integral towers of rank r. Weight functions with values in N^r and the corresponding weighted total-degree monomial orderings lift naturally from one domain R_j-1in the tower to the next, R_j, the integral closure of R_j-1[x_j]/<0(x_j)>. The...

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 Gröbner bases.

We present a variant of the Diffie-Hellman scheme in which the number of bits exchanged is one third of what is used in the classical Diffie-Hellman scheme, while the offered security against attacks known today is the same. We also give applications for this variant and conjecture a extension of this variant further reducing the size of sent infor...

Improved geometric Goppa codes have a smaller redundancy and the same bound on the minimum distance as ordinary algebraic-geometry codes. For an asymptotically good sequence of function fields we give a formula for the redundancy

Coding theory deals with the following topics: Cryptography or cryptology. Transmission of secret messages or electronic money, eavesdropping, intruders, authentication and privacy. Source coding or data compression. Most data have redundant information, and can be compressed, to save space or to speed up the transmission. Error-correcting codes. I...

From the previous chapter one might get the impression that the theory of error-correcting codes is equivalent to the theory of finite geometry or arrangements over finite fields. This is not true from a practical point of view. A code is useless without a decoding algorithm. For engineers the total performance of the encoding and decoding scheme i...

In this project we give examples of methods described in the Chapters 10 and 11 on finding the minimum weight codewords, the decoding of cyclic codes and working with the Mathieu groups (see also 6). The codes that we use here are the well-known Golay codes. These codes are among the most beautiful objects in coding theory, and we would like to giv...

The Weierstrass semigroups of some places in an asymptotically good tower of function fields are computed.

The authors discuss a method to get plane curves with many rational points and a construction to get asymptotically good codes and curves, that is closely related to finding bivariate polynomials representing designs. The number of rational points and the genus is computed for plane curves that have a defining equation with three monomials

The order bound on generalized Hamming weights is introduced in a general setting of codes on varieties which comprises both the one point geometric Goppa codes as well as the q-ary Reed-Muller codes. For the latter codes it is shown that this bound is sharp and that they satisfy the double chain condition

The projective plane curve with defining equation X 3Y + Y 3Z + Z 3X = 0 has been studied for numerous reasons since Klein [19].

Based on the notion of an order function we construct and determine the parameters of a class of error-correcting evaluation codes. This class includes the one-point algebraic geometry codes as well as the general- ized Reed-Muller codes, and the parameters are determined without using heavy machinery from algebraic geometry.

We study minimal blocking sets in PG(2,q) having q + m points outside some fixed line. If 0 < m < (

We consider an asymptotically good tower (Ym)m 1 of curves over a finite field Fq, where q is a square. The Weierstrass semigroup Hm of a rational point P (m) 1 of Ym is determined. For n 2 Hm and m = 1,2 and 3 we will give rational functions on Ym that have pole order n at P (m) 1 and that have no poles outside P (m) 1 . For larger m an explicit d...

Algebraic-geometric codes have a t-error-correcting pair which corrects errors up to half the designed minimum distance. A generalization of the Roos bound is given from cyclic to linear codes. An MDS code of minimum distance 5 has a 2-error-correcting pair if and only if it is an extended-generalized-Reed-Solomon code.

We give a generalization of the shift bound on the minimum distance for cyclic codes which applies to Reed-Muller and algebraic-geometric codes. The number of errors one can correct by majority coset decod- ing is up to half the shift bound.

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 the...

This paper provides a survey of the existing literature on the decoding of algebraic-geometric codes. Definitions, theorems, and cross references will be given. We show what has been done, discuss what still has to be done, and pose some open problems

We give necessary and sufficient conditions for two geometric Goppa codes C L (D, G) and C L (D, H) to be the same. As an application we characterize self-dual geometric Goppa codes.

Abstract This talk is intended to give a survey on the existing literature on the decoding of algebraic-geometric codes. Although the motivation originally was to find an ecient decoding algorithm for algebraic-geometric codes, the latest results give algorithms which can be explained purely in terms of linear algebra. We will treat the following s...

Decoding geometric Goppa codes can be reduced to solving the key congruence of a received word in an affine ring. If the codelength is smaller than the number of rational points on the curve, then this method can correct up to 1.2 ( d *-L)/2- s errors, where d * is the designed minimum distance of the code and s is the Clifford defect. The affine...

We generalize the existing decoding algorithms by error location for BCH and algebraic-geometric codes to arbitrary linear codes. We investigate the number of dependent sets of error positions. A received word with an independent set of error positions can be corrected.

In this paper, a necessary and sufficient criterion for self-duality of geometric Goppa codes is given. Resumé On donne une condition necessaire et suffisante pour l'autodualité d'un code géométrique de Goppa.

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If one imposes conditions on the degree of the divisor used, then we derive criteria for linear codes to be algebraic-geometric. In particular, we investigate the family of q-ary Hamming codes, and prove that on...

A decoding algorithm for algebraic geometric codes that was given by A.N. Skorobogatov and S.G. Vladut (preprint, Inst. Problems of Information Transmission, 1988) is considered. The author gives a modified algorithm, with improved performance, which he obtains by applying the above algorithm a number of times in parallel. He proves the existence o...

This paper is a report on the ongoing research concerning the extended coset leader weight enumerator using the theory of arrangements of hyperplanes, geo-metric lattices and characteristic polynomials.

This talk is intended to give a survey on the existing literature on the decoding of algebraic-geometric codes. Although the motivation originally was to find an ecient decoding algorithm for algebraic-geometric codes, the latest results give algorithms which can be explained purely in terms of linear algebra. We will treat the following subjects:

The q-ary Reed-Muller codes RMq(u;m) of length n = qm are a general- ization of Reed-Solomon codes, which use polynomials in m variables to encode messages through functional encoding. Using an idea of reducing the multivariate case to the uni- variate case, randomized list-decoding algorithms for Reed-Muller codes were given in (1) and (15). The a...

Abstract The gonality sequence of a plane curve is computed. A two variable zeta function for curves over a,nite,eld is dened,and the rationality and a functional equation are proved. 1991 Mathematics Subject Classication: 14G10, 94B27.

The decoding of arbitrary linear block codes is accomplished by solving a system of quadratic equations by means of Buchberger's algorithm for finding a Gröbner basis. This generalizes the algorithm of Berlekamp-Massey for decoding Reed-Solomon, Goppa and cyclic codes up to half the true minimum distance by intro-ducing the unknown syndromes as var...