Unbounded Error Correcting Codes

We introduce a variant of Error Correcting Codes with no predetermined length. An Unbounded ECC with rate R and distance ε is an encoding of a possibly infinite message into a possibly infinite codeword, such that for every large enough k we may recover the first Rk symbols of the message from the first k symbols of the codeword – even when up to 1/2ε k of these codeword symbols are adversarially corrupted. We study unbounded codes over a binary alphabet in the regime of small distance ε, and obtain nearly-tight upper and lower bounds in several natural settings. We show that the optimal rate of such a code is between R<1-Ω(√(ε)) and R>1-O(√(εloglog(1/ε))). Surprisingly, our construction is non-linear, and we show that the optimal rate of a linear unbounded code is the asymptotically worse R=1-Θ(√(εlog(1/ε))). In the setting of random noise, the optimal rate of unbounded codes improves and matches the rate of standard codes at R=1-Θ(εlog(1/ε)).