Efficiently find the permutation guaranteed by the probabilistic existence theorem

Develop an efficient algorithm to find the permutation of the blocks in a covering design that is guaranteed by Theorem 4.9 to yield a CFR code with zero skip cost.

Background

Theorem 4.9 establishes non-constructively that, for fixed integers 2 ≤ t < k and sufficiently large v, any (t,k,v) covering design can be arranged as the columns of a CFR code with locality ⌈k/(t−1)⌉ and zero skip cost. The proof uses a probabilistic argument to show that a suitable independent permutation of the elements in each block exists.

The theorem does not provide an efficient procedure for identifying the required permutations. The authors explicitly identify finding such a permutation efficiently as the most important open question, because an algorithmic construction would convert the existential result into a practical method for producing optimal-expansion zero-skip-cost CFR codes.