7 research outputs found
Optimal k-Deletion Correcting Codes
Levenshtein introduced the problem of constructing k-deletion correcting codes in 1966, proved that the optimal redundancy
of those codes is O(k log N), and proposed an optimal redundancy single-deletion correcting code (using the so-called VT
construction). However, the problem of constructing optimal redundancy k-deletion correcting codes remained open. Our key
contribution is a solution to this longstanding open problem. We present a k-deletion correcting code that has redundancy 8k log n+
o(log n) and encoding/decoding algorithms of complexity O(n^(2k+1)) for constant k
Locally Decodable Codes with Randomized Encoding
We initiate a study of locally decodable codes with randomized encoding.
Standard locally decodable codes are error correcting codes with a
deterministic encoding function and a randomized decoding function, such that
any desired message bit can be recovered with good probability by querying only
a small number of positions in the corrupted codeword. This allows one to
recover any message bit very efficiently in sub-linear or even logarithmic
time. Besides this straightforward application, locally decodable codes have
also found many other applications such as private information retrieval,
secure multiparty computation, and average-case complexity.
However, despite extensive research, the tradeoff between the rate of the
code and the number of queries is somewhat disappointing. For example, the best
known constructions still need super-polynomially long codeword length even
with a logarithmic number of queries, and need a polynomial number of queries
to achieve a constant rate. In this paper, we show that by using a randomized
encoding, in several models we can achieve significantly better rate-query
tradeoff. In addition, our codes work for both the standard Hamming errors, and
the more general and harder edit errors.Comment: 23 page