An Improved Bound on the Fraction of Correctable Deletions

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