3,179 research outputs found
A Note on Cyclic Codes from APN Functions
Cyclic codes, as linear block error-correcting codes in coding theory, play a
vital role and have wide applications. Ding in \cite{D} constructed a number of
classes of cyclic codes from almost perfect nonlinear (APN) functions and
planar functions over finite fields and presented ten open problems on cyclic
codes from highly nonlinear functions. In this paper, we consider two open
problems involving the inverse APN functions and the Dobbertin
APN function . From the calculation of
linear spans and the minimal polynomials of two sequences generated by these
two classes of APN functions, the dimensions of the corresponding cyclic codes
are determined and lower bounds on the minimum weight of these cyclic codes are
presented. Actually, we present a framework for the minimal polynomial and
linear span of the sequence defined by ,
where is a primitive element in . These techniques can also be
applied into other open problems in \cite{D}
A Novel Stochastic Decoding of LDPC Codes with Quantitative Guarantees
Low-density parity-check codes, a class of capacity-approaching linear codes,
are particularly recognized for their efficient decoding scheme. The decoding
scheme, known as the sum-product, is an iterative algorithm consisting of
passing messages between variable and check nodes of the factor graph. The
sum-product algorithm is fully parallelizable, owing to the fact that all
messages can be update concurrently. However, since it requires extensive
number of highly interconnected wires, the fully-parallel implementation of the
sum-product on chips is exceedingly challenging. Stochastic decoding
algorithms, which exchange binary messages, are of great interest for
mitigating this challenge and have been the focus of extensive research over
the past decade. They significantly reduce the required wiring and
computational complexity of the message-passing algorithm. Even though
stochastic decoders have been shown extremely effective in practice, the
theoretical aspect and understanding of such algorithms remains limited at
large. Our main objective in this paper is to address this issue. We first
propose a novel algorithm referred to as the Markov based stochastic decoding.
Then, we provide concrete quantitative guarantees on its performance for
tree-structured as well as general factor graphs. More specifically, we provide
upper-bounds on the first and second moments of the error, illustrating that
the proposed algorithm is an asymptotically consistent estimate of the
sum-product algorithm. We also validate our theoretical predictions with
experimental results, showing we achieve comparable performance to other
practical stochastic decoders.Comment: This paper has been submitted to IEEE Transactions on Information
Theory on May 24th 201
On Some Properties of Quadratic APN Functions of a Special Form
In a recent paper, it is shown that functions of the form
, where and are linear, are a good source for
construction of new infinite families of APN functions. In the present work we
study necessary and sufficient conditions for such functions to be APN
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