75,137 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
Linear Codes from Some 2-Designs
A classical method of constructing a linear code over \gf(q) with a
-design is to use the incidence matrix of the -design as a generator
matrix over \gf(q) of the code. This approach has been extensively
investigated in the literature. In this paper, a different method of
constructing linear codes using specific classes of -designs is studied, and
linear codes with a few weights are obtained from almost difference sets,
difference sets, and a type of -designs associated to semibent functions.
Two families of the codes obtained in this paper are optimal. The linear codes
presented in this paper have applications in secret sharing and authentication
schemes, in addition to their applications in consumer electronics,
communication and data storage systems. A coding-theory approach to the
characterisation of highly nonlinear Boolean functions is presented
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