Amortized Relaxed Locally Decodable Codes

RESUMO

Locally decodable codes (LDCs) enable recovery of any message symbol by probing only a small number of positions in a possibly corrupted codeword. The central parameters of an LDC are its rate, locality, and error tolerance. Ideally, one would like all three parameters to be constant. However, classical lower bounds show that such codes cannot exist. A recent line of work introduced amortized locally decodable codes (aLDCs), in which the decoder is tasked with recovering an entire block of consecutive message symbols rather than a single symbol. While prior work obtained ideal aLDCs with constant rate, constant error tolerance, and constant amortized locality, those constructions relied on either shared randomness hidden from the channel or computational assumptions restricting the channel. Another well-studied relaxation is the notion of a relaxed locally decodable code (RLDC), in which the decoder may output a special failure symbol $\bot$ rather than risk decoding incorrectly. In this work, we introduce the notion of an amortized relaxed locally decodable code (aRLDC), combining amortized decoding with the relaxed decoding paradigm. Unlike prior ideal aLDC constructions, our model is fully information-theoretic and makes no assumptions about shared randomness or computational limitations of the adversarial channel. We construct the first aRLDC with constant rate, constant error tolerance, and constant amortized locality. Moreover, for any block of length $\Omega(\mathrm{polylog}(k))$, our decoder achieves amortized locality $1+\delta^{1 - o(1)}$, where $\delta$ is the error tolerance parameter. Thus, asymptotically, recovering a long block requires essentially less than two codeword probe per message symbol recovered. By contrast, without amortization no RLDC can simultaneously achieve constant rate, constant error tolerance, and constant locality.