122 research outputs found
Maximum distance separable 2D convolutional codes
Maximum distance separable (MDS) block codes and MDS 1D convolutional codes are the most robust codes for error correction within the class of block codes of a fixed rate and 1D convolutional codes of a certain rate and degree, respectively. In this paper, we generalize this concept to the class of 2D convolutional codes. For that, we introduce a natural bound on the distance of a 2D convolutional code of rate and degree , which generalizes the Singleton bound for block codes and the generalized Singleton bound for 1D convolutional codes. Then, we prove the existence of 2D convolutional codes of rate and degree that reach such bound when if , or if , by presenting a concrete constructive procedure
Cyclone Codes
We introduce Cyclone codes which are rateless erasure resilient codes. They
combine Pair codes with Luby Transform (LT) codes by computing a code symbol
from a random set of data symbols using bitwise XOR and cyclic shift
operations. The number of data symbols is chosen according to the Robust
Soliton distribution. XOR and cyclic shift operations establish a unitary
commutative ring if data symbols have a length of bits, for some prime
number . We consider the graph given by code symbols combining two data
symbols. If such random pairs are given for data symbols, then a
giant component appears, which can be resolved in linear time. We can extend
Cyclone codes to data symbols of arbitrary even length, provided the Goldbach
conjecture holds.
Applying results for this giant component, it follows that Cyclone codes have
the same encoding and decoding time complexity as LT codes, while the overhead
is upper-bounded by those of LT codes. Simulations indicate that Cyclone codes
significantly decreases the overhead of extra coding symbols
Lifted MDS Codes over Finite Fields
MDS codes are elegant constructions in coding theory and have mode important
applications in cryptography, network coding, distributed data storage,
communication systems et. In this study, a method is given which MDS codes are
lifted to a higher finite field. The presented method satisfies the protection
of the distance and creating the MDS code over the by using MDS code over
$F_p.
Hankel Rhotrices and Constructions of Maximum Distance Separable Rhotrices over Finite Fields
Many block ciphers in cryptography use Maximum Distance Separable (MDS) matrices to strengthen the diffusion layer. Rhotrices are represented by coupled matrices. Therefore, use of rhotrices in the cryptographic ciphers doubled the security of the cryptosystem. We define Hankel rhotrix and further construct the maximum distance separable rhotrices over finite fields
Lectures on Designing Screening Experiments
Designing Screening Experiments (DSE) is a class of information - theoretical
models for multiple - access channels (MAC). We discuss the combinatorial model
of DSE called a disjunct channel model. This model is the most important for
applications and closely connected with the superimposed code concept. We give
a detailed survey of lower and upper bounds on the rate of superimposed codes.
The best known constructions of superimposed codes are considered in paper. We
also discuss the development of these codes (non-adaptive pooling designs)
intended for the clone - library screening problem. We obtain lower and upper
bounds on the rate of binary codes for the combinatorial model of DSE called an
adder channel model. We also consider the concept of universal decoding for the
probabilistic DSE model called a symmetric model of DSE.Comment: 66 page
Twisted Reed-Solomon Codes
We present a new general construction of MDS codes over a finite field
. We describe two explicit subclasses which contain new MDS codes
of length at least for all values of . Moreover, we show that
most of the new codes are not equivalent to a Reed-Solomon code.Comment: 5 pages, accepted at IEEE International Symposium on Information
Theory 201
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