I have recently completed a survey “Codes with local decoding procedures” and will be giving a talk on this matter in August at the ICM. The survey covers two related families of codes with locality, namely, locally decodable codes that are broadly useful in theoretical computer science and local reconstruction codes that have recently been used in data storage applications to provide a more space efficient alternative to RAID. Below is the abstract of the survey:
Error correcting codes allow senders to add redundancy to messages, encoding bit strings representing messages into longer bit strings called codewords, in a way that the message can still be recovered even if a fraction of the codeword bits are corrupted. In certain settings however the receiver might not be interested in recovering all the message, but rather seek to quickly recover just a few coordinates of it. Codes that allow one to recover individual message coordinates extremely fast (locally), from accessing just a small number of carefully chosen coordinates of a corrupted codeword are said to admit a local decoding procedure. Such codes have recently played an important role in several areas of theoretical computer science and have also been used in practice to provide reliability in large distributed storage systems. We survey what is known about these codes.