  We study sheaf codes, a type of linear codes with a fixed hierarchical
collection of local codes, viewed as a sheaf of vector spaces on a finite
topological space we call coded space. Many existing codes, such as tensor
product codes, Sipser-Spielman codes, and their more recent high-dimensional
analogs, can be naturally represented as sheaf codes on simplicial and cubical
complexes, considered as coded spaces. We introduce a new property of a sheaf
code, called maximal extendibility, which ensures that within a class of codes
on the same coded space, we encounter as few obstructions as possible when
extending local sections globally. We show that in every class of sheaf codes
defined on the same space and parameterized by parity-check matrices with
polynomial entries, there always exists a maximally extendable sheaf code. Such
codes are very interesting since it is possible to show that maximally
extendable tensor product codes are good coboundary expanders, which
potentially could be used to attack the qLTC conjecture.
