Are there good locally decodable _erasure_ codes Are there known locally decodable erasure codes with
  
  linear codeword length and $\:n^{o(1)}\:$ query complexity?


According to pages 1 and 4 of this link (which annoying does not give its own date,

though it cites references from 2011), locally decodable (error-correcting) codes

with linear codeword length are only known with query complexity $n^{\epsilon}$ for $\: \epsilon &gt; 0 \;$.

i.e., Does considering erasure codes instead change that?

Such a code could be used for showing that a data host has random access to a given file.
Locally decodable codes and locally decodable erasure codes are qualitatively equivalent. Both imply $\Omega(m)$ many disjoint $q$-tuples from which one can recover a given message coordinate, where $m$ is the codeword length and $q$ is the query complexity. A formal argument appears in Section 3.4 of Kerenidis and de Wolf's paper.