Close the existential gap for binary k-deletion codes

Close the existential gap between the known lower and upper bounds for the maximum size D(n,k) of a binary length-n k-deletion code when k is at least 2, improving the lower bound beyond the current Omega_k(2^n\log n/n^{2k}) scale toward the upper bound O_k(2^n/n^k).

Background

For a binary length-n k-deletion code, D(n,k) denotes the maximum number of codewords such that no two codewords share a common subsequence of length n-k. The paper summarizes the known bounds as an upper bound of O_k(2^n/n^k) and, for k at least 2, an improved lower bound of approximately 2^n\log n/n^{2k}. Thus, for k at least 2, there remains a polynomial gap of roughly n^k, up to logarithmic factors, between the existential lower and upper bounds.

The paper focuses instead on list-decodable deletion codes and proves new lower bounds for those objects; it does not resolve the stated gap for ordinary deletion codes. Consequently, determining the optimal order of D(n,k) for k at least 2 remains an explicit unresolved problem.