Asymptotic unit rate for the quadratic Reed–Muller storage construction

Determine whether the full-parity storage-code rates associated with the quadratic Reed–Muller evaluation construction tend to one as the input dimension tends to infinity; equivalently, establish whether the binary rank of the indicator matrix indexed by symmetric binary matrices, with entries indicating that the sum of two matrices has rank at most one, is little-o of the number of symmetric binary matrices.

Background

The quadratic Reed–Muller construction uses the evaluation map sending a binary vector x to the symmetric rank-one matrix xx^T. Its associated Cayley graph has the space of symmetric binary matrices as its vertex set, with two vertices adjacent when their sum has matrix rank one. The full-parity storage rate is determined by the binary rank of the matrix whose entries indicate whether the sum of two symmetric matrices has rank at most one.

The paper obtains a nontrivial asymptotic lower bound on this storage rate, but does not prove convergence to one. The unresolved question is therefore whether the relevant binary rank is subexponential relative to the graph order, or equivalently whether the storage deficiency tends to zero.