On a Problem of Persi Diaconis

Abstract

A problem of Persi Diaconis is to construct a cyclic sequence s₁s₂,…, s_nwhose elements lie in the m-dimensional binary vector space, such that every d-dimensional subspace is spanned by precisely one subsequence of lengthd. Because the cyclic sequence has n subsequences of lengthdn must be the number of d-dimensional subspaces of the m-dimensional binary vector space, which is $\frac{{\left( {2^m - 1} \right)\left( {2^m - 2} \right) \ldots \left( {2^m - 2^{d - 1} } \right)}} {{\left( {2^d - 1} \right)\left( {2^d - 2} \right) \ldots \left( {2^d - 2^{d - 1} } \right)}}$ This paper solves the case ofd = 2.