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We consider data transmission over a network where each edge is an erasure channel and the inner nodes transmit the random linear combinations of their incoming information. We distinguish two channel models in this setting: the row and the column erasure channel model. For both models, we investigate spread codes and determine their symbol erasure correction capability and the probability of decoding success. We also compare the performance of spread codes to other known codes suitable for those models. Furthermore, we explain how to decode these codes in the two channel models and compare the decoding complexities. The results show that depending on the application and the to-be-optimized aspect, any combination of codes and channel models can be the best choice. [ABSTRACT FROM AUTHOR]

We would like to thank Mahdieh Hakimi Poroch and Ali Asghar Talebi for pointing out two errors in Proposition 28 and Theorem 29 of the original paper [1]. The correct formulation for these two statements is as follows. [ABSTRACT FROM PUBLISHER]

Private information retrieval (PIR) schemes for coded storage with colluding servers are presented, 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. In the special case of no server collusion, this rate coincides with the known asymptotic PIR capacity for MDS-coded storage systems. 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. [ABSTRACT FROM AUTHOR]