4,709 research outputs found
Relaxation Bounds on the Minimum Pseudo-Weight of Linear Block Codes
Just as the Hamming weight spectrum of a linear block code sheds light on the
performance of a maximum likelihood decoder, the pseudo-weight spectrum
provides insight into the performance of a linear programming decoder. Using
properties of polyhedral cones, we find the pseudo-weight spectrum of some
short codes. We also present two general lower bounds on the minimum
pseudo-weight. The first bound is based on the column weight of the
parity-check matrix. The second bound is computed by solving an optimization
problem. In some cases, this bound is more tractable to compute than previously
known bounds and thus can be applied to longer codes.Comment: To appear in the proceedings of the 2005 IEEE International Symposium
on Information Theory, Adelaide, Australia, September 4-9, 200
On the Minimal Pseudo-Codewords of Codes from Finite Geometries
In order to understand the performance of a code under maximum-likelihood
(ML) decoding, it is crucial to know the minimal codewords. In the context of
linear programming (LP) decoding, it turns out to be necessary to know the
minimal pseudo-codewords. This paper studies the minimal codewords and minimal
pseudo-codewords of some families of codes derived from projective and
Euclidean planes. Although our numerical results are only for codes of very
modest length, they suggest that these code families exhibit an interesting
property. Namely, all minimal pseudo-codewords that are not multiples of a
minimal codeword have an AWGNC pseudo-weight that is strictly larger than the
minimum Hamming weight of the code. This observation has positive consequences
not only for LP decoding but also for iterative decoding.Comment: To appear in Proc. 2005 IEEE International Symposium on Information
Theory, Adelaide, Australia, September 4-9, 200
Mathematical Programming Decoding of Binary Linear Codes: Theory and Algorithms
Mathematical programming is a branch of applied mathematics and has recently
been used to derive new decoding approaches, challenging established but often
heuristic algorithms based on iterative message passing. Concepts from
mathematical programming used in the context of decoding include linear,
integer, and nonlinear programming, network flows, notions of duality as well
as matroid and polyhedral theory. This survey article reviews and categorizes
decoding methods based on mathematical programming approaches for binary linear
codes over binary-input memoryless symmetric channels.Comment: 17 pages, submitted to the IEEE Transactions on Information Theory.
Published July 201
On Pseudocodewords and Improved Union Bound of Linear Programming Decoding of HDPC Codes
In this paper, we present an improved union bound on the Linear Programming
(LP) decoding performance of the binary linear codes transmitted over an
additive white Gaussian noise channels. The bounding technique is based on the
second-order of Bonferroni-type inequality in probability theory, and it is
minimized by Prim's minimum spanning tree algorithm. The bound calculation
needs the fundamental cone generators of a given parity-check matrix rather
than only their weight spectrum, but involves relatively low computational
complexity. It is targeted to high-density parity-check codes, where the number
of their generators is extremely large and these generators are spread densely
in the Euclidean space. We explore the generator density and make a comparison
between different parity-check matrix representations. That density effects on
the improvement of the proposed bound over the conventional LP union bound. The
paper also presents a complete pseudo-weight distribution of the fundamental
cone generators for the BCH[31,21,5] code
A note on the stability number of an orthogonality graph
We consider the orthogonality graph Omega(n) with 2^n vertices corresponding
to the 0-1 n-vectors, two vertices adjacent if and only if the Hamming distance
between them is n/2. We show that the stability number of Omega(16) is
alpha(Omega(16))= 2304, thus proving a conjecture by Galliard. The main tool we
employ is a recent semidefinite programming relaxation for minimal distance
binary codes due to Schrijver.
As well, we give a general condition for Delsarte bound on the (co)cliques in
graphs of relations of association schemes to coincide with the ratio bound,
and use it to show that for Omega(n) the latter two bounds are equal to 2^n/n.Comment: 10 pages, LaTeX, 1 figure, companion Matlab code. Misc. misprints
fixed and references update
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