1,587 research outputs found
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
Low-degree tests at large distances
We define tests of boolean functions which distinguish between linear (or
quadratic) polynomials, and functions which are very far, in an appropriate
sense, from these polynomials. The tests have optimal or nearly optimal
trade-offs between soundness and the number of queries.
In particular, we show that functions with small Gowers uniformity norms
behave ``randomly'' with respect to hypergraph linearity tests.
A central step in our analysis of quadraticity tests is the proof of an
inverse theorem for the third Gowers uniformity norm of boolean functions.
The last result has also a coding theory application. It is possible to
estimate efficiently the distance from the second-order Reed-Muller code on
inputs lying far beyond its list-decoding radius
Results on the Generalized Covering Radius of Error Correcting Codes
The recently proposed generalized covering radius is a fundamental property of error correcting codes. This quantity characterizes the trade off between time and space complexity of certain algorithms when a code is used in them. However, for the most part very little is known about the generalized covering radius. My thesis seeks to expand on this field in several ways. First, a new upper bound on this quantity is established and compared to previous bounds. Second, this bound is used to derive a new algorithm for finding codewords within the generalized covering radius of a given vector, and also to modify an existing algorithm, greatly improving its efficiency
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