Compressing sparse sequences under local decodability constraints

We consider a variable-length source coding problem subject to local decodability constraints. In particular, we investigate the blocklength scaling behavior attainable by encodings of r-sparse binary sequences, under the constraint that any source bit can be correctly decoded upon probing at most d codeword bits. We consider both adaptive and non-adaptive access models, and derive upper and lower bounds that often coincide up to constant factors. Notably, such a characterization for the fixed-blocklength analog of our problem remains unknown, despite considerable research efforts. Connections to communication complexity are also briefly discussed.