Two Sparse Binary Operators with Optimal Generating Length

We consider two specified rotation-and-shift operators on a 32-dimensional binary vector space. Their unital algebra is the full matrix algebra, and their generating length is exactly ten, the smallest possible value for any pair of 32-by-32 matrix generators. A certificate supplies 1,024 independent words of length at most ten. A second proof gives an explicit polynomial rank-one matrix and two cyclic bases. These bases construct every matrix by a unique coefficient array. The conclusions hold over every field of characteristic two. The paper includes exact certificates, independent verification, and the consequences for simultaneous invariant subspaces and linear quotients.