text: 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.
  @cite_14: Private information retrieval (PIR) is the problem of retrieving as efficiently as possible, one out of @math messages from @math non-communicating replicated databases (each holds all @math messages) while keeping the identity of the desired message index a secret from each individual database. The information theoretic capacity of PIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. @math -private PIR is a generalization of PIR to include the requirement that even if any @math of the @math databases collude, the identity of the retrieved message remains completely unknown to them. Robust PIR is another generalization that refers to the scenario where we have @math databases, out of which any @math may fail to respond. For @math messages and @math databases out of which at least some @math must respond, we show that the capacity of @math -private and Robust PIR is @math . The result includes as special cases the capacity of PIR without robustness ( @math ) or @math -privacy constraints ( @math ).
  @cite_4: In the private information retrieval (PIR) problem, a user wishes to retrieve, as efficiently as possible, one out of @math messages from @math non-communicating databases (each holds all @math messages) while revealing nothing about the identity of the desired message index to any individual database. The information theoretic capacity of PIR is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. For @math messages and @math databases, we show that the PIR capacity is @math . A remarkable feature of the capacity achieving scheme is that if we eliminate any subset of messages (by setting the message symbols to zero), the resulting scheme also achieves the PIR capacity for the remaining subset of messages.
  @cite_12: We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of @math non-colluding databases, each storing a coded version of @math messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is @math , where @math is the rate of the @math code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost. The result generalizes the achievability and converse results for the classical PIR with replicating databases to the case of coded databases.
summary: While explicit PIR schemes which achieve capacity are constructed in @cite_4 , @cite_14 , and @cite_12 , they require the base field to be large. If @math is the number of servers and @math is the number of files, the capacity-achieving schemes of @cite_14 require a field size of @math , since they rely on the existence of MDS codes of high lengths. Realistic storage systems, however, may operate over fields of small size to keep the complexity of the involved operations manageable. One would naturally then like to construct explicit PIR schemes over small base fields.