2,658 research outputs found
Covering Radius of the -rd Order Reed-Muller Code in the Set of Resilient Functions
In this paper, we continue the study of the covering
radius in the set of resilient functions, which has been defined by Kurosawa. This new concept is meaningful to cryptography especially in the context of the new class of algebraic attacks on stream ciphers proposed by Courtois and Meier at Eurocrypt 2003 and Courtois at Crypto 2003. In order to resist such attacks the combining Boolean function should be at high distance from lower degree functions.
Using a result from coding theory on the covering radius of
-rd Reed-Muller codes, we establish exact values of the
the covering radius of in the set of -resilient Boolean
functions of variables, when . We also improve the lower bounds for covering radius of the Reed-Muller
codes in the set of -resilient functions, where
, and
On the Peak-to-Mean Envelope Power Ratio of Phase-Shifted Binary Codes
The peak-to-mean envelope power ratio (PMEPR) of a code employed in
orthogonal frequency-division multiplexing (OFDM) systems can be reduced by
permuting its coordinates and by rotating each coordinate by a fixed phase
shift. Motivated by some previous designs of phase shifts using suboptimal
methods, the following question is considered in this paper. For a given binary
code, how much PMEPR reduction can be achieved when the phase shifts are taken
from a 2^h-ary phase-shift keying (2^h-PSK) constellation? A lower bound on the
achievable PMEPR is established, which is related to the covering radius of the
binary code. Generally speaking, the achievable region of the PMEPR shrinks as
the covering radius of the binary code decreases. The bound is then applied to
some well understood codes, including nonredundant BPSK signaling, BCH codes
and their duals, Reed-Muller codes, and convolutional codes. It is demonstrated
that most (presumably not optimal) phase-shift designs from the literature
attain or approach our bound.Comment: minor revisions, accepted for IEEE Trans. Commun
New Set of Codes for the Maximum-Likelihood Decoding Problem
The maximum-likelihood decoding problem is known to be NP-hard for general
linear and Reed-Solomon codes. In this paper, we introduce the notion of
A-covered codes, that is, codes that can be decoded through a polynomial time
algorithm A whose decoding bound is beyond the covering radius. For these
codes, we show that the maximum-likelihood decoding problem is reachable in
polynomial time in the code parameters. Focusing on bi- nary BCH codes, we were
able to find several examples of A-covered codes, including two codes for which
the maximum-likelihood decoding problem can be solved in quasi-quadratic time.Comment: in Yet Another Conference on Cryptography, Porquerolle : France
(2010
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