Update-efficiency and local repairability limits for capacity approaching codes

Motivated by distributed storage applications, we investigate the degree to which capacity achieving codes can be efficiently updated when a single information symbol changes, and the degree to which such codes can be efficiently repaired when a single encoded symbol is lost. Specifically, we first develop conditions under which optimum error-correction and update-efficiency are possible. We establish that the number of encoded bits that should change in response to a change in a single information bit must scale logarithmically in the block-length of the code, if we are to achieve any nontrivial rate with vanishing probability of error over the binary erasure or binary symmetric channels. Moreover, we show that there exist capacity-achieving codes with this scaling. With respect to local repairability, we develop tight upper and lower bounds on the number of remaining encoded bits that are needed to recover a single lost encoded bit. In particular, we show that when the rate of an optimal code is ε below capacity, the maximum number of codeword symbols required to recover one lost symbol must scale as log1/ε. Several variations on - and extensions of - these results are also developed, including to the problem of rate-distortion coding.