Lower Bound on the Redundancy of PIR Codes

We prove that the redundancy of a k-server PIR code of dimension s is Ω( √ s) for all k > 3. This coincides with a known upper bound of O( √ s) on the redundancy of PIR codes. The same lower bound was proved independently by Mary Wootters [3] using a different method. Given two binary vectors u = (u1, u2, . . . , un) and v = (v1, v2, . . . , vn), we define their product uv componentwise, namely uv def = (u1v1, u2v2, . . . , unvn) (1) where u1v1, u2v2, . . . , unvn are computed in GF(2). Note that the product operation in (1) distributes over addition in Fn 2 . Thus (1) turns the vector space F n 2 into an algebra An over F2. This algebra An is unital, associative, and commutative. Given a set X ⊆ Fn 2 , we define the square of X as the set of products of the elements in X. Explicitly, X is defined as follows: X def = { uv : u, v ∈ X } (2) The following lemmas follow straightforwardly from the definitions in (1) and (2), along with the fact that An is a commutative algebra. We let 〈X〉 denote the linear span over F2 of a set X ⊆ Fn 2 . Lemma 1. |X2| 6 |X| ( |X|+ 1 ) /2. Proof. If |X| = r, then X consists of the ( 2 ) vectors uv = vu for some u 6= v in X, along with the r vectors uu = u for some u ∈ X. Some of these vectors may coincide. Lemma 2. If a, b ∈〈X〉, then ab ∈ 〈 X 〉 . Proof. Let X = {x1, x2, . . . , xr}. Write a = ∑i αixi and b = ∑i βixi for some binary coefficients α1, α2, . . . αr, β1β2, . . . , βr. Then