On linear-combinatorial problems associated with subspaces spanned by $\{\pm 1\}$-vectors

Abstract

A complete answer to the question about subspaces generated by $\{\pm 1\}$-vectors, which arose in the work of I.Kanter and H.Sompolinsky on associative memories, is given. More precisely, let vectors $v_1,\dots ,v_p,$ $p\le n-1,$ be chosen at random uniformly and independently from $\{\pm 1{\}}^n\subset {\mathbf{R}}^n.$ Then the probability $\mathbb{P}(p,n)$ that $$span⟨v_1,\dots ,v_p⟩\cap \left\{\{\pm 1{\}}^n\setminus \{\pm v_1,\dots ,\pm v_p\}\right\}\ne \varnothing$$ is shown to be $$4{p \choose 3}\left(\frac{3}{4}\right)^n + O\left(\left(\frac{5}{8} + o_n(1)\right)^n\right) \quad \mbox{as} \quad n\to \infty,$$ where the constant implied by the $O$-notation does not depend on $p$. The main term in this estimate is the probability that some 3 vectors $v_{j_1},v_{j_2},v_{j_3}$ of $v_j$, $j=1,\dots ,p,$ have a linear combination that is a $\{\pm 1\}$-vector different from $\pm v_{j_1},\pm v_{j_2},\pm v_{j_3}.$

13 pages