Local computation in codes

A locally testable code allows one to store vast amounts of data, where estimating the fraction of errors in the data takes roughly as much time as takes to read one bit of the data! If the fraction of errors is below a certain threshold, a locally decodeable code would allow one to recover every bit of the original message, again, in time which is roughly the time to read one bit of the data. Are such locally testable/decodeable codes of constant rate possible? So far we don´t know, but surprisingly-good codes are known. Following, we survey some of the literature and discuss a connection between these notions to symmetric LDPC codes.