An Improved Bound on the Fraction of Correctable Deletions

We consider codes over fixed alphabets against worst case symbol deletions. For any fixed k ≥ 2, we construct a family of codes over alphabet of size k with positive rate, which allow efficient recovery from a worst case deletion fraction approaching 1 - (2/(k + √k)). In particular, for binary codes, we are able to recover a fraction of deletions approaching 1/(√2+1) = √2-1 ≈ 0.414. Previously, even non-constructively, the largest deletion fraction known to be correctable with √ positive rate was 1 - θ(1/ k), and around 0.17 for the binary case. Our result pins down the largest fraction of correctable deletions for k-ary codes as 1 - Θ(1/k), since 1 - 1/k is an upper bound even for the simpler model of erasures where the locations of the missing symbols are known. Closing the gap between (√2 - 1) and 1/2 for the limit of worst case deletions correctable by binary codes remains a tantalizing open question.