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.

翻译：我们研究层丛码（sheaf codes）——一种具有固定分层局部码集合的线性码，可视为在有限拓扑空间（称为编码空间）上的向量空间层。许多现有编码，如张量积码、Sipser-Spielman码及其近期的高维类比，均可自然地表示为单纯复形和立方复形（作为编码空间）上的层丛码。我们引入层丛码的一个新性质——最大可扩展性，该性质保证：在同一编码空间上的同类编码中，将局部截面全局扩展时遇到的障碍尽可能少。我们证明：在定义于同一空间、由多项式元素校验矩阵参数化的所有层丛码类中，总存在最大可扩展层丛码。这类编码具有重要意义，因为可以证明最大可扩展张量积码是优质的共边界扩展开，这有望用于解决qLTC猜想。