The enumeration of certain sets of block maps (Q1112821)

An n-block is a sequence \(b_ 1,b_ 2...b_ n\), where \(b_ i\in Z_ 2\) for \(1\leq i\leq n\), and an n-block map is a function from the set of n- blocks to the ring \(Z_ 2\). The n-block maps form the ring \(GF(2)\{x_ 1,x_ 2,...,x_ n| x^ 2_ i=1\}\). The set of all block maps with the operations of addition and polynomial substitution form a near-ring. The general problem of searching nonlinear block maps for factors with respect to polynomial substitution seems to be extremely complex. However, effective search procedures have been described for factors within certain sets of block maps which are nonlinear but are linear in the first variable. In this paper we show that the smallest of these sets contains just over 85 \% of the block maps which are linear in the first variable. The largest of the sets, for which there is an additional step in the search procedure, contains over 99 \% of the block maps which are linear in the first variable.