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.
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