337 research outputs found
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
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
The Hamiltonian problem and -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 -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 -path traceable graphs to characterizing
those that have no universal vertex. We generalize a construction of maximal
non-traceable graphs by Zelinka to -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 . 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
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