List Decoding of Hermitian Codes using Groebner Bases

List decoding of Hermitian codes is reformulated to allow an efficient and simple algorithm for the interpolation step. The algorithm is developed using the theory of Groebner 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.Comment: 19 pages, 2 figure

Algebraic Soft-Decision Decoding of Hermitian Codes

An algebraic soft-decision decoder for Hermitian codes is presented. We apply Koetter and Vardy's soft-decision decoding framework, now well established for Reed-Solomon codes, to Hermitian codes. First we provide an algebraic foundation for soft-decision decoding. Then we present an interpolation algorithm finding 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-Solomon codes, favorable to the former.Comment: 17 pages, submitted to IEEE Transaction on Information Theor

The Hamiltonian problem and tt-path traceable graphs

The problem of characterizing maximal non-Hamiltonian graphs may be naturally extended to characterizing graphs that are maximal with respect to non-traceability and beyond that to $t$-path traceability. We show how traceability behaves with respect to disjoint union of graphs and the join with a complete graph. Our main result is a decomposition theorem that reduces the problem of characterizing maximal $t$-path traceable graphs to characterizing those that have no universal vertex. We generalize a construction of maximal non-traceable graphs by Zelinka to $t$-path traceable graphs.Comment: 12 pages, 4 figure

List Decoding of Reed-Solomon Codes from a Groebner Basis Perspective

The interpolation step of Guruswami and Sudan's 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 Groebner bases of modules. In a special case, this algorithm reduces to a simple Berlekamp-Massey-like decoding algorithm.Comment: submitted to the Journal of Symbolic Computation; Shortened and revised versio

On Semigroups Generated by Two Consecutive Integers and Improved Hermitian Codes

Analysis of the Berlekamp-Massey-Sakata algorithm for decoding one-point codes leads to two methods for improving code rate. One method, due to Feng and Rao, removes parity checks that may be recovered by their majority voting algorithm. The second method is to design the code to correct only those error vectors of a given weight that are also geometrically generic. In this work, formulae are given for the redundancies of Hermitian codes optimized with respect to these criteria as well as the formula for the order bound on the minimum distance. The results proceed from an analysis of numerical semigroups generated by two consecutive integers. The formula for the redundancy of optimal Hermitian codes correcting a given number of errors answers an open question stated by Pellikaan and Torres in 1999.Comment: Added reference

The Berlekamp-Massey Algorithm and the Euclidean Algorithm: a Closer Link

The two primary decoding algorithms for Reed-Solomon codes are the Berlekamp-Massey algorithm and the Sugiyama et al. adaptation of the Euclidean algorithm, both designed to solve a key equation. In this article an alternative version of the key equation and a new way to use the Euclidean algorithm to solve it are presented, which yield the Berlekamp-Massey algorithm. This results in a new, simpler, and compacter presentation of the Berlekamp-Massey algorithm

Duality for Several Families of Evaluation Codes

We consider generalizations of Reed-Muller codes, toric codes, and codes from certain plane curves, such as those defined by norm and trace functions on finite fields. In each case we are interested in codes defined by evaluating arbitrary subsets of monomials, and in identifying when the dual codes are also obtained by evaluating monomials. We then move to the context of order domain theory, in which the subsets of monomials can be chosen to optimize decoding performance using the Berlekamp-Massey-Sakata algorithm with majority voting. We show that for the codes under consideration these subsets are well-behaved and the dual codes are also defined by monomials.Comment: Added reference

The Sum-Product Algorithm for Degree-2 Check Nodes and Trapping Sets

The sum-product algorithm for decoding of binary codes is analyzed for bipartite graphs in which the check nodes all have degree $2$. The algorithm simplifies dramatically and may be expressed using linear algebra. Exact results about the convergence of the algorithm are derived and applied to trapping sets.Comment: Unpublished, 26 page

Redundancies of Correction-Capability-Optimized Reed-Muller Codes

This article is focused on some variations of Reed-Muller codes that yield improvements to the rate for a prescribed decoding performance under the Berlekamp-Massey-Sakata algorithm with majority voting. Explicit formulas for the redundancies of the new codes are given

The Order Bound on the Minimum Distance of the One-Point Codes Associated to a Garcia-Stichtenoth Tower of Function Fields

Garcia and Stichtenoth discovered two towers of function fields that meet the Drinfeld-Vl\u{a}du\c{t} bound on the ratio of the number of points to the genus. For one of these towers, Garcia, Pellikaan and Torres derived a recursive description of the Weierstrass semigroups associated to a tower of points on the associated curves. In this article, a non-recursive description of the semigroups is given and from this the enumeration of each of the semigroups is derived as well as its inverse. This enables us to find an explicit formula for the order (Feng-Rao) bound on the minimum distance of the associated one-point codes.Comment: A new section is added in which the order bound is derive