Consider a linear [n, k, d] q code C. We say that that i-th coordinate of C has locality r, if the value at this coordinate can be recovered from accessing some other r coordinates of C. Data storage applications require codes with small redundancy, low locality for information coordinates, large distance, and low locality for parity coordinates. In this paper we carry out an in-depth study of the relations between these parameters.We establish a tight bound for the redundancy n − k in terms of the message length, the distance, and the locality of information coordinates. We refer to codes attaining the bound as optimal. We prove some structure theorems about optimal codes, which are particularly strong for small distances. This gives a fairly complete picture of the tradeoffs between codewords length, worst-case distance and locality of information symbols.We then consider the locality of parity check symbols and erasure correction beyond worst case distance for optimal codes. Using our structure theorem, we obtain a tight bound for the locality of parity symbols possible in such codes for a broad class of parameter settings. We prove that there is a tradeoff between having good locality for parity checks and the ability to correct erasures beyond the minimum distance.

Synthetic DNA is durable and can encode digital data with high density, making it an attractive medium for data storage. However, recovering stored data on a large-scale currently requires all the DNA in a pool to be sequenced, even if only a subset of the information needs to be extracted. Here, we encode and store 35 distinct files (over 200 MB of data), in more than 13 million DNA oligonucleotides, and show that we can recover each file individually and with no errors, using a random access approach. We design and validate a large library of primers that enable individual recovery of all files stored within the DNA. We also develop an algorithm that greatly reduces the sequencing read coverage required for error-free decoding by maximizing information from all sequence reads. These advances demonstrate a viable, large-scale system for DNA data storage and retrieval.

Consider a systematic linear code where some (local) parity symbols depend on few prescribed symbols, while other (heavy) parity symbols may depend on all data symbols. Local parities allow to quickly recover any single symbol when it is erased, while heavy parities provide tolerance to a large number of simultaneous erasures. A code as above is maximally-recoverable, if it corrects all erasure patterns which are information theoretically recoverable given the code topology. In this paper we present explicit families of maximally-recoverable codes with locality. We also initiate the study of the trade-off between maximal recoverability and alphabet size.Definition 3. Let C be a data-local (k, r, h)-code. We say that C is maximally-recoverable if for any set E ⊆ [n], where E is obtained by picking one coordinate from each of k r local groups, puncturing C in coordinates in E yields a maximum distance separable [k + h, k] code.A [k+h, k] MDS code obviously corrects all patterns of h erasures. Therefore a maximally-recoverable data-local (k, r, h)-code corrects all erasure patterns E ⊆ [n] that involve erasing one coordinate per local group, and h additional coordinates. We now argue that any erasure pattern that is not dominated by a pattern above has to be uncorrectable.Lemma 4. Let C be an arbitrary data-local (k, r, h)-code. Let E ⊆ [n] be an erasure pattern. Suppose E affects t local groups and |E| > t + h; then E is not correctable.

No abstract