Margreta Kuijper

On constructing a shortest linear recurrence relation

It has been shown in the literature that a formulation of the minimal partial realization problem in terms of exact modeling of a behavior lends itself to an iterative polynomial solution. For the scalar case, we explicitly present such a solution in full detail. Unlike classical solution methods based on Hankel matrices, the algorithm is constructive. It iteratively constructs a partial realization of minimal McMillian degree. The algorithm is known in information theory as the Berlekamp-Massey algorithm and is used for constructing a shortest linear recurrence relation for a finite sequence of numbers.

On Minimality of Convolutional Ring Encoders

Convolutional codes are considered with code sequences modeled as semi-infinite Laurent series. It is well known that a convolutional code C over a finite group G has a minimal trellis representation that can be derived from code sequences. It is also well known that, for the case that G is a finite field, any polynomial encoder of C can be algebraically manipulated to yield a minimal polynomial encoder whose controller canonical realization is a minimal trellis. In this paper we seek to extend this result to the finite ring case G = \BBZpr by introducing a so-called ldquo p-encoderrdquo. We show how to manipulate a polynomial encoding scheme of a noncatastrophic convolutional code over \BBZpr to produce a particular type of p-encoder (ldquominimal p -encoderrdquo) whose controller canonical realization is a minimal trellis with nonlinear features. The minimum number of trellis states is then expressed as p gamma, where gamma is the sum of the row degrees of the minimal p -encoder. In particular, we show that any convolutional code over \BBZpr admits a delay-free p -encoder which implies the novel result that delay-freeness is not a property of the code but of the encoder, just as in the field case. We conjecture that a similar result holds with respect to catastrophicity, i.e., any catastrophic convolutional code over \BBZpr admits a noncatastrophic p-encoder.

A Parametric Approach to List Decoding of Reed-Solomon Codes Using Interpolation

In this paper, we present a minimal list decoding algorithm for Reed-Solomon (RS) codes. Minimal list decoding for a code C refers to list decoding with radius L, where L is the minimum of the distances between the received word r and any codeword in C. We consider the problem of determining the value of L as well as determining all the codewords at distance L. Our approach involves a parametrization of interpolating polynomials of a minimal Gröbner basis G . We present two efficient ways to compute G. We also show that so-called re-encoding can be used to further reduce the complexity. We then demonstrate how our parametric approach can be solved by a computationally feasible rational curve fitting solution from a recent paper by Wu. Besides, we present an algorithm to compute the minimum multiplicity as well as the optimal values of the parameters associated with this multiplicity, which results in overall savings in both memory and computation.

Descriptor representations without direct feedthrough term

Abstract Descriptor representations are considered that are given by ( E , A , B , C , D ) with D = 0. Minimality under external equivalence is characterized in terms of the matrices E , A , B and C . Also, transformations are given by which minimal ( E , A , B , C ) representations are related under external equivalence. The transformations turn out to be more simple than in the “ D ≠ 0” case. Algorithms for rewriting an ( E , A , B , C , D ) representation in ( E , A , B , C ) form are also given. Finally, a realization procedure is presented for obtaining a minimal ( E , A , B , C ) representation for a system that is given in polynomial matrix fractional form.

BEHAVIORAL MODELS OVER RINGS MINIMAL REPRESENTATIONS AND APPLICATIONS TO CODING AND SEQUENCES

Abstract In this paper we consider polynomial kernel representations for behaviors. For behaviors over fields it is well-known that minimal representations, i.e. representations with minimal row degrees, are exactly those representations for which the polynomial matrix is row reduced. In this paper we consider behaviors over a particular type of ring, namely ℤ p r , where p is a prime number and r is a positive integer. As a starting point in this investigation we focus on minimal partial realizations. These are equivalent to shortest linear recurrence relations. We present an algorithm that computes a parametrization of all shortest linear recurrence relations for a finite sequence in ℤ p r . For this we extend well-known techniques developed by Reeds and Sloane in the 80s with methods from the theory of behavioral modeling.

Source coding with side information using list decoding

Existing literature on source coding with side information (SCSI) uses channel codes like LDPC codes and turbo codes and assumes classical unique decoding. In this paper, in contrast to classical decoding, we have taken the list decoding approach and show that the theoretical limit of SCSI can then be achieved. We argue that, as opposed to channel coding, the correct sequence from the list produced by the list decoder can effectively be recovered in case of SCSI with a few CRC bits. The CRC bits, which allow the decoder to identify the correct sequence, incur negligible overhead for large block length. More importantly, these CRC bits are not subject to noise since we are dealing with a virtual noisy channel rather than a real noisy channel. Finally, we present a guideline for designing constructive SCSI schemes using Reed Solomon code, BCH code, and Reed-Muller code, which are the known list-decodable codes.

Behavioral models for list decoding

Recently it has been shown that list decoding of Reed-Solomon codes may be translated into a bivariate interpolation problem. The data consist of pairs in a finite field and the aim is to find a bivariate polynomial that interpolates the given pairs and is minimal with respect to some criterion. We present a systems theoretic approach to this interpolation problem. With the data points we associate a set of time series, also called trajectories. For this set of trajectories we construct the Most Powerful Unfalsified Model (MPUM). This is the smallest possible model that explains these trajectories. The bivariate polynomial is then derived from a specific polynomial representation of the MPUM.

A distributed source coding framework for multiple sources

In this paper the problem of compressing multiple correlated sources is addressed. It is assumed that the m sources are given as a sequence where an assumption is made only on the correlation between neighbouring sources. The correlation assumption is deterministic and Hamming-distance based (so-called ldquoconstrained correlationrdquo in the literature). The problem of compressing such a sequence of correlated sources is easily solved by repeatedly applying the asymmetric distributed source coding (DSC) for two sources, as in the DISCUS framework of (S.S. Pradhan et al., 2003). In this paper we present an alternative, more general, framework that achieves the same overall compression rate but allows for more flexible compression rates per source. The proposed encoder employs a channel code with a parity check matrix that does not need to be systematic. The proposed decoder consists of three steps. Firstly, difference patterns between sources are recovered from the compressed data via a channel decoder. Subsequently, this information is combined with other parts of the compressed data. Finally, by solving a linear equation and adding difference patterns, the original messages are reconstructed. We show that, for m = 2, several DSC coding schemes from the literature are special cases. We illustrate our framework for a sequence of 3 sources through a (15, 7) BCH code with a non-systematic parity check matrix.

Gröbner bases and behaviors over finite rings

For several decades Gröbner bases have proved useful tools for different areas in system theory, particularly multidimensional system theory. These areas range from controller design to minimal realizations of linear systems over fields. In this paper we focus on the univariate case and identify the so-called “predictable leading monomial property” as a property of a minimal Gröbner basis that is crucial in many of these areas. The property is stronger than “row reducedness”. We revisit the recently developed theory of [17] in which row reducedness is extended to polynomial matrices over the finite ring ℤ<inf>p</inf>^r (with p a prime integer and r a positive integer), which find applications in error control coding over ℤ<inf>p</inf>^r. We recast the ideas of [17] in the more general setting of Gröbner bases and derive new results on how to use minimal Gröbner bases to achieve the predictable leading monomial property over ℤ<inf>p</inf>^r. A major advantage of the Gröbner approach is that computational packages are available to compute a minimal Gröbner basis over ℤ<inf>p</inf>^r, such as the SINGULAR computer algebra system. Another advantage of the Gröbner approach is its generality with respect to the choice of ordering of polynomial vectors.

Distributed source coding with cyclic codes and their duals

We consider the problem of compression of multiple correlated sources. It is assumed that the sources are given as a sequence with neighbouring sources correlated in terms of Hamming distance. In previous work we derived a general Distributed Source Coding (DSC) framework that uses the parity check matrix of a suitably chosen block code C1. In this paper we focus on the special case that C1 is a cyclic code. This allows us to formulate our DSC framework in terms of polynomials. The polynomial structure has the advantage that it allows for implementation via efficient shift registers. We show that the Sylvester resultant of polynomials plays a crucial role in the DSC framework. We address the notion of “complementarity” of codes and specify the DSC framework in terms of the cyclic code C1 and the reciprocal C2 of its dual code for the special case that C2 is complementary to C1. We illustrate our polynomial DSC framework for a sequence of 3 sources through a (15, 7) BCH code C1 and the reciprocal of its dual code C2.