Reed-Solomon cubical sheaf code [1–3]

Description

DLV code on the degree-\(k\) cochains of a sheaf with projective Reed-Solomon local codes over an arithmetic quotient of a product of \(r\) Bruhat-Tits trees. Its parameters are chosen so that Ramanujan expansion and uniform product expansion of the local codes yield binary QLTCs with linear distance and constant soundness [1,3].

The edges at a vertex in direction \(i\) form a projective line over the residue field \(\mathbb{F}_{q_i}\). The local code evaluates homogeneous polynomials on that line, with one degree per endpoint color [1; Sec. 12][3; Lemma 4.4]. A Frobenius reindexing at one endpoint color makes the local codes compatible with the arithmetic identifications, so they glue into a sheaf on the quotient [1,3]. Cochains take values in a characteristic-two field containing every \(\mathbb{F}_{q_i^2}\), and restriction of scalars to \(\mathbb{F}_2\) yields the qubit code [1; Sec. 2.3][3; Thm. 6.2].

In Ref. [3], the quotient comes from a quaternionic lattice over \(\mathbb{F}_2(y)\) [4,5]. Ref. [2] assumed uniform product expansion of its local codes, which Ref. [3; Sec. 1.3 and Thm. 5.4] proves for its parameters.