Constant-additive redundancy for general k-substring edit correction

Determine whether the additive O_{q,k}(\log\log n) redundancy for general q-ary k-substring edit correcting codes can be reduced to O_{q,k}(1), thereby achieving redundancy \log n+O_{q,k}(1) for fixed q and k.

Background

The paper constructs q-ary k-substring edit correcting codes with redundancy \log n+8\log\log n+o(\log\log n) for fixed alphabet size q and edit parameter k. It also establishes the lower bound \log n-O_{q,k}(1), leaving an additive O_{q,k}(\log\log n) gap between the construction and the information-theoretic lower bound.

The authors note that their unequal-length construction achieves \log n+O_{q,k}(1) redundancy, but the same approach does not directly extend to equal-length edits: Proposition B.1 shows that a straightforward extension fails for a particular syndrome class. Consequently, new techniques are required to determine whether constant-additive redundancy is attainable for the full k-substring edit model.