Michael E. O'Sullivan

Algebraic construction of sparse matrices with large girth

In this correspondence, we present a method for constructing sparse matrices that have a compact description and whose associated bipartite graphs have large girth. Based on an arbitrary seed matrix of nonnegative integers a new matrix is constructed which replaces each entry of the seed matrix with a sum of permutation matrices. Algebraic conditions that lead to short cycles in the associated bipartite graph are analyzed and methods to achieve large girth in two special cases are presented. In one, all the permutation matrices are circulants; in the other they are all affine permutation matrices. When used to define a low-density parity-check (LDPC) code the compact description should lead to efficient implementation and the large girth to good error correction performance. The method is adaptable to a variety of rates, and a variety of row and column degrees

List decoding of Reed-Solomon codes from a Gröbner basis perspective

The interpolation step of Guruswami and Sudans list decoding of Reed-Solomon codes poses the problem of finding the minimal polynomial of an ideal with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Grobner bases of modules. In a special case, this algorithm reduces to a simple Berlekamp-Massey-like decoding algorithm.

An Interpolation Algorithm using Gröbner Bases for Soft-Decision Decoding of Reed-Solomon Codes

A central problem of algebraic soft-decision decoding of Reed-Solomon codes is to find the minimal polynomial of the ideal of interpolating polynomials with respect to a certain monomial order. An efficient algorithm that solves the problem is presented based on the theory of Grobner bases of modules

The linear programming bound for codes over finite Frobenius rings

In traditional algebraic coding theory the linear-programming bound is one of the most powerful and restrictive bounds for the existence of both linear and non-linear codes. This article develops a linear-programming bound for block codes on finite Frobenius rings.

List decoding of Hermitian codes using Gröbner bases

List decoding of Hermitian codes is reformulated to yield an efficient and simple algorithm for the interpolation step. The algorithm is developed using the theory of Grobner bases of modules. The computational complexity of the algorithm seems comparable to previously known algorithms achieving the same task, and the algorithm is better suited for hardware implementation.

Algebraic Soft-Decision Decoding of Hermitian Codes

An algebraic soft-decision decoder for Hermitian codes is presented. We apply Koetter and Vardys soft-decision decoding framework, now well established for Reed-Solmon codes, to Hermitian codes. First we provide an algebraic foundation for soft-decision decoding. Then we present an interpolation algorithm to find the Q-polynomial that plays a key role in the decoding. With some simulation results, we compare performances of the algebraic soft-decision decoders for Hermitian codes and Reed-Solmon codes, favorable to the former.

Failure prediction and diagnosis for satellite monitoring systems using Bayesian networks

Predicting failure in complex systems, such as satellite network systems, is a challenging problem. A satellite earth terminal contains many components, including high-powered amplifiers, signal converters, modems, routers, and generators, any of which may cause system failure. The ability to estimate accurately the probability of failure of any of these components, given the current state of the system, may help reduce the cost of operation. Probabilistic graphical models, in particular Bayesian networks, provide a consistent framework in which to address problems containing uncertainty and complexity. Building a Bayesian network for failure prediction in a complex system such as a satellite earth terminal requires a large quantity of data. Software monitoring systems have the potential to provide vast amounts of data related to the operating state of the satellite earth terminal. Measurable nodes of the Bayesian network correspond to states of measurable parameters in the system and unmeasurable nodes represent failure of various components. Nodes for environmental factors are also included. A description of Bayesian networks will be provided and a demonstration of inference on the Bayesian network, such as the calculation of the marginal probability of failure nodes given measurements and the maximum probability state of the system for failure diagnosis will be given. Using the data to learn local probabilities of the network will be covered. An interface between MaxView monitoring and control software and a Bayesian network API will also be described.

Unique Decoding of Plane AG Codes via Interpolation

We present a unique decoding algorithm of algebraic geometry (AG) codes on plane curves, Hermitian codes in particular, from an interpolation point of view. The algorithm successfully corrects errors of weight up to half of the order bound on the minimum distance of the AG code. It is the first decoding algorithm to combine some features of the interpolation-based list decoding with the performance of the syndrome decoding with the majority voting scheme. The regular structure of the algorithm allows a straightforward parallel implementation.

ON PLOTKIN-OPTIMAL CODES OVER FINITE FROBENIUS RINGS

We study the Plotkin bound for codes over a finite Frobenius ring R equipped with the homogeneous weight. We show that for codes meeting the Plotkin bound, the distribution on R induced by projection onto a coordinate has an interesting property. We present several constructions of codes meeting the Plotkin bound and of Plotkin-optimal codes. We also investigate the relationship between Butson–Hadamard matrices and codes over R meeting the Plotkin bound.

The key equation for one-point codes and efficient error evaluation ☆

Abstract In this article, the key equation and the use of error evaluator polynomials are generalized from the case of BCH codes to one-point codes. We interpret the syndrome of the error vector e as a differential ω e which has simple poles on the support of e and, in general, at the one-point Q used to define the codes. The decoding problem is to find a function f and differential φ having poles only at Q such that fω e = φ . Then if f has a simple pole at an error position P , the error value is e P =( φ /d f )( P ). We amend an iterative algorithm that computes a Grobner basis for I e , the ideal of functions vanishing on the support of e , so that it also computes the corresponding error evaluators. That is, we produce fω e for each f in the Grobner basis.