Locally Decodable Codes for Edit Distance

Abstract

Locally decodable codes (LDC) [1,9] are error correcting codes that allow decoding (any) individual symbol of the message, by reading only few symbols of the codeword. LDC’s, originally considered in the setting of PCP’s [1], have found other additional applications in theory of CS, such as PIR in cryptography, generating a lot of fascinating work (see [12] and references within). In one straightforward practical application to storage, such codes provide enormous efficiency gains over standard error correcting codes (ECCs), that need to read the entire encoded message to learn even a single bit of the encoded message. Typically, LDC’s, as well as standard ECC’s are designed to decode the encoded message if up to some bounded fraction of the symbols had been modified. This corresponds to decoding strings of bounded Hamming distance from a valid codeword. A stronger natural metric is the edit distance, measuring the shortest sequence of insertions and deletions (indel.) of symbols leading from one word to another. Standard ECC’s for edit distance have been previously considered [11]. Furthermore, [11] devised codes with rate and distance (error tolerance) optimal up to constants, with efficient encoding and decoding procedures. However, combining these two useful settings of LDC, and robustness against indel. errors has never been considered.

In this work, we study the question of constructing LDC’s for edit distance. We demonstrate a strong positive result - LDC’s for edit distance can be achieved, with similar parameters to LDC’s for Hamming distance. More precisely, we devise a generic transformation from LDC for Hamming distance to LDC for edit distance with related parameters. Besides the theoretical appeal of such a primitive, we put forward a cryptographic application to secure property testing over storage prone to indel. errors (such as DNA-based storage).