This paper presents private information retrieval (PIR) schemes for coded storage with colluding servers, which are not restricted to maximum distance separable (MDS) codes. PIR schemes for general linear codes are constructed and the resulting PIR rate is calculated explicitly. It is shown that codes with transitive automorphism groups yield the highest possible rates obtainable with the proposed scheme. This rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems without collusion. While many PIR schemes in the literature require field sizes that grow with the number of servers and files in the system, we focus especially on the case of a binary base field, for which Reed- Muller codes serve as an important and explicit class of examples.
In this work we construct PIR schemes based on general linear codes, and concentrate in particular on binary Reed--Muller (RM) codes. The schemes described in @cite_8 employed Generalized Reed-Solomon (GRS) codes, and the resulting analysis of the achievable rate relied on the @math of two GRS codes @math and @math again being a GRS code. The class of RM codes is closed under the star product operation as well, and thus naturally lends itself to be employed using the PIR scheme of @cite_8 . However, RM codes have the advantage of being defined over the binary field @math . When comparing GRS and RM codes of equal length and dimension, it is shown here that the same PIR rates as with GRS codes can be achieved in the non-colluding case. For a fixed PIR rate, however, RM codes provide less protection against collusion due to their lower minimum distance. Nevertheless, it is shown that the @math -PIR RM schemes presented here still provide protection against a substantial fraction of colluding sets of sizes slightly bigger than @math . In more detail, the main contributions of this paper are:
@cite_8
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The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi [1] who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with rephcated-server PIR. In the present paper, the construction of an (n, k) τ-server binary linear PIR code having parameters n = lΣ i=0 (m i ), k = (m i ) and τ = 2l for any integer m ≥ l ≥ 0 is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary. Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any τ = ∊ 2l, 2l − 1 | l ∊ Z, l ≥ 0 and any value of k, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with known short block length codes in [1]. For the particular case of τ = 3,4, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length n k, τ) of a systematic PIR code. It follows from a result by Vardy and Yaakobi [2], that these codes also yield optimal, systematic primitive multi-set n, k, τ) B batch codes for τ = 3,4. The PIR code constructions presented here also yield upper bounds on the generahzed Hamming weights of binary PRM codes.