Maximally Extendable Sheaf Codes
Pavel Panteleev, Gleb Kalachev
TL;DR
The paper develops a unifying framework of sheaf codes on finite coded spaces to study local-to-global consistency in linear codes. It proves that in any class of sheaf codes with polynomially parameterized local parity checks, maximally extendable (ME) codes exist over sufficiently large finite fields, enabling robust global extendability of local sections. This ME property yields strong coboundary-expansion behavior for multi-dimensional product codes, offering a pathway toward constructing good qLTCs via higher-dimensional topological structures. The work also develops a broad toolkit of operations, cohomology, and expansion notions for sheaf codes, and suggests future directions including Ramanujan cubical complexes and Tanner/color-based quantum codes to push toward practical quantum LDPC code constructions.
Abstract
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.