Linear codes over finite extension fields have widespread applications in
theory and practice. In some scenarios, the decoder has a sequential access to
the codeword symbols, giving rise to a hierarchical erasure structure. In this
paper we develop a mathematical framework for hierarchical erasures over
extension fields, provide several bounds and constructions, and discuss
potential applications in distributed storage and flash memories. Our results
show intimate connection to Universally Decodable Matrices, as well as to
Reed-Solomon and Gabidulin codes.