This is an announcement for the paper "Polynomial bounds for large Bernoulli sections of $\ell_1^N$" by S. Artstein-Avidan, O. Friedland, V. Milman, and S. Sodin. Abstract: We prove a quantitative version of the bound on the smallest singular value of a Bernoulli covariance matrix (due to Bai and Yin). Then we use this bound, together with several recent developments, to show that the distance from a random (1-delta) n - dimensional section of l_1^n, realised as an image of a sign matrix, to an Euclidean ball is polynomial in 1/delta (and independent of n), with high probability. Archive classification: Functional Analysis; Metric Geometry; Mathematical Physics Remarks: 22 pages The source file(s), polyl13.tex: 38003 bytes, is(are) stored in gzipped form as 0601369.gz with size 13kb. The corresponding postcript file has gzipped size 68kb. Submitted from: sodinale@post.tau.ac.il The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/math.FA/0601369 or http://arXiv.org/abs/math.FA/0601369 or by email in unzipped form by transmitting an empty message with subject line uget 0601369 or in gzipped form by using subject line get 0601369 to: math@arXiv.org.