A Fixed-Length Generation Phenomenon in Hypercubes

This note establishes a fixed-length generation phenomenon for the cycle space of the hypercube. We prove a result for the n-dimensional hypercube: for every integer m>=2 and n>=m, the entire cycle space is generated by simple cycles of length 2m. In the proof, each square face is expressed as the symmetric difference of two simple cycles of length 2m. In short, sufficiently high-dimensional hypercubes admit a uniform fixed-length generating family for their cycle spaces.