For q-ary sequences, t-deletion reconstruction codes need only O(1) redundancy at N around n^{t-1}, while t-insertion codes need log log n, and new two-error codes achieve 3 log n down to log n redundancy for N=2 through 5.
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Reconstruction Codes for Deletions and Insertions: Connection, Distinction, and Construction
For q-ary sequences, t-deletion reconstruction codes need only O(1) redundancy at N around n^{t-1}, while t-insertion codes need log log n, and new two-error codes achieve 3 log n down to log n redundancy for N=2 through 5.