3-query locally decodable codes of subexponential length

Locally Decodable Codes (LDC) allow one to decode any particular symbol of the input message by making a constant number of queries to a codeword, even if a constant fraction of the codeword is damaged. In a recent work [Yek08] Yekhanin constructs a 3-query LDC with sub-exponential length of size exp(exp(O( log n log log n ))). However, this construction requires a conjecture that there are infinitely many Mersenne primes. In this paper we give an unconditional 3-query LDC construction with a shorter codeword length of exp(exp(O( log n log log n))). We also give a 2r -query LDC with length of exp(exp(O( r log n log logr-1 n))). The main ingredient in our construction is the existence of super-polynomial size set-systems with restricted intersections by [Gro00] which hold only over composite numbers.

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Klim Efremenko
• Keywords: Algorithms, Log Log N, Log Logr-1 N, Shorter Codeword Length, STOC 2009