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 − (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 − (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.
Together with Håstad REF they later improved the deletion fraction to 1 − 2 q+ √ q or √ 2 − 1 ≈ 0.414 for binary codes.
An Improved Bound on the Fraction of Correctable Deletions
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Mathematics Computer Science