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 1−2k+1. In particular, for binary codes, we are able to recover a fraction of deletions approaching 1/3. 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 1/3 and 1/2 for the limit of worst-case deletions correctable by binary codes remains a tantalizing open question.