1. New Locally Decodable Codes and Private Information Retrieval Schemes
Sergey Yekhanin

2. LDCs and PIRs
Definition: A code C encoding n bits to N bits is called q LDC if every bit of the message can be recovered (w.h.p.) by a randomized decoder reading only q bits of the encoding even after some constant fraction of the encoding has been corrupted.
Definition: A q server PIR protocol is a protocol between a user and q non-communicating servers holding an n bit database D, that allows the user to retrieve any bit Di, while leaking no information about i to any server.

3. LDCs: progress q Lower bound Upper bound 1 Do not exist [KT] 2
Exp(n) [KdW]
Exp(n) [Folklore] 3
Ω(n3/2) [KT]
Ω(n2/log2 n) [KdW]
Ω(n2/log n) [W]
Exp(n1/2) [BIK]
Exp(n1/32,582,657) [Y]
Exp(nO(1/log log n)) [Y]
Exp(nO(1/log1-ε log n)) [Y]

4. PIRs: progress q Lower bound Upper bound 1 Θ(n) [CGKS] 2 5 log n [WdW]
O(n1/3) [CGKS] 3
O(n1/5) [A]
O(n1/5.25) [BIKR]
O(n1/32,582,658) [Y]
O(nO(1/log log n)) [Y]
O(nO(1/log1-ε log n)) [Y]

5. Proof overview Goal: 3 query LDCs of length Exp(n1/3)
Regular Intersecting Families (RIFs)
RIFs yield LDCs
Basic linear-algebraic construction of RIFs
Combinatorial and algebraic niceness of sets
Main construction of RIFs