Smallest order for graphs with independent exact r-covers when d equals 7

Determine the smallest order of a 7-regular simple graph that has an independent exact r-cover for every integer r with 1 ≤ r ≤ 7, resolving the remaining smallest-degree case identified by the paper.

Background

The paper defines N(d) as the minimum order of a d-regular simple graph possessing an independent exact r-cover for every r ≤ d. It establishes optimal constructions for d = 3, 4, 5, and 6, but reports that d = 7 remains unresolved. For d = 7, the known lower and upper bounds differ by a factor of 2, so determining the exact minimum requires either a sharper lower bound or a construction meeting the existing lower bound.