## DIMACS TR: 99-06

## A New Upper Bound on the
Reliability Function of the Gaussian Channel

### Authors: A. Ashikhmin, A. Barg, and S. Litsyn

**
ABSTRACT
**

We derive a new upper bound on the
exponent of error probability of decoding for the best possible
codes in the Gaussian channel. This bound is tighter than the
known upper bounds (the sphere-packing and minimum-distance bounds
proved in Shannon's classical 1959 paper and their low-rate
improvement by Kabatiansky and Levenshtein).
The proof is accomplished by studying asymptotic
properties of codes on the Euclidean $n$-dimensional sphere.
First we prove that the distance distribution of codes of large
size necessarily contains a large component.
A general theorem establishing this estimate is proved simultaneously
for codes on the Euclidean sphere and in real and complex projective
spaces.

To derive specific estimates of the distance distribution, we study
the asymptotic behavior of Jacobi polynomials $P_k^{\alpha,\beta}$ as
$k\to \infty$ and at least one of the upper indices grows linearly in
$k$. This group of results provides the exact behavior of the exponent
of Jacobi polynomials in the entire orthogonality segment.

Since on the average there are many code vectors in the vicinity of
the transmitted vector $\bfx$, one can show that the probability of
confusing $\bfx$ and one of these vectors cannot be too small. This
proves a lower bound on the error probability of decoding and the
upper bound announced in the title.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1999/99-06.ps.gz

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