3,792 research outputs found
Second-Order Weight Distributions
A fundamental property of codes, the second-order weight distribution, is
proposed to solve the problems such as computing second moments of weight
distributions of linear code ensembles. A series of results, parallel to those
for weight distributions, is established for second-order weight distributions.
In particular, an analogue of MacWilliams identities is proved. The
second-order weight distributions of regular LDPC code ensembles are then
computed. As easy consequences, the second moments of weight distributions of
regular LDPC code ensembles are obtained. Furthermore, the application of
second-order weight distributions in random coding approach is discussed. The
second-order weight distributions of the ensembles generated by a so-called
2-good random generator or parity-check matrix are computed, where a 2-good
random matrix is a kind of generalization of the uniformly distributed random
matrix over a finite filed and is very useful for solving problems that involve
pairwise or triple-wise properties of sequences. It is shown that the 2-good
property is reflected in the second-order weight distribution, which thus plays
a fundamental role in some well-known problems in coding theory and
combinatorics. An example of linear intersecting codes is finally provided to
illustrate this fact.Comment: 10 pages, accepted for publication in IEEE Transactions on
Information Theory, May 201
A New Class of MDS Erasure Codes Based on Graphs
Maximum distance separable (MDS) array codes are XOR-based optimal erasure
codes that are particularly suitable for use in disk arrays. This paper
develops an innovative method to build MDS array codes from an elegant class of
nested graphs, termed \textit{complete-graph-of-rings (CGR)}. We discuss a
systematic and concrete way to transfer these graphs to array codes, unveil an
interesting relation between the proposed map and the renowned perfect
1-factorization, and show that the proposed CGR codes subsume B-codes as their
"contracted" codes. These new codes, termed \textit{CGR codes}, and their dual
codes are simple to describe, and require minimal encoding and decoding
complexity.Comment: in Proceeding of IEEE Global Communications Conference (GLOBECOM
Lattices from Codes for Harnessing Interference: An Overview and Generalizations
In this paper, using compute-and-forward as an example, we provide an
overview of constructions of lattices from codes that possess the right
algebraic structures for harnessing interference. This includes Construction A,
Construction D, and Construction (previously called product
construction) recently proposed by the authors. We then discuss two
generalizations where the first one is a general construction of lattices named
Construction subsuming the above three constructions as special cases
and the second one is to go beyond principal ideal domains and build lattices
over algebraic integers
Weight Distributions of Regular Low-Density Parity-Check Codes over Finite Fields
The average weight distribution of a regular low-density parity-check (LDPC)
code ensemble over a finite field is thoroughly analyzed. In particular, a
precise asymptotic approximation of the average weight distribution is derived
for the small-weight case, and a series of fundamental qualitative properties
of the asymptotic growth rate of the average weight distribution are proved.
Based on this analysis, a general result, including all previous results as
special cases, is established for the minimum distance of individual codes in a
regular LDPC code ensemble.Comment: 15 pages, 5 figures, accepted for publication in IEEE Transactions on
Information Theory, July 201
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