  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 over the last three
decades. Connections to communication complexity are also briefly discussed.
