Let \(C\) and \(D\) be chain complexes over \(\mathbb {F}_2\). The graded object underlying the tensor product double complex is defined, for integers \(p, q \in \mathbb {Z}\), by

where \(\otimes _{\mathbb {F}_2}\) denotes the tensor product of \(\mathbb {F}_2\)-modules.

Let \(C\) and \(D\) be chain complexes over \(\mathbb {F}_2\). The horizontal differential

is defined by

that is, it applies the differential \(\partial ^C\) of \(C\) in the first (horizontal, \(p\)-)direction and the identity on \(D_q\).

Let \(C\) and \(D\) be chain complexes over \(\mathbb {F}_2\). The vertical differential

is defined by

that is, it applies the identity on \(C_p\) and the differential \(\partial ^D\) of \(D\) in the second (vertical, \(q\)-)direction.

Let \(C\) and \(D\) be chain complexes over \(\mathbb {F}_2\). The tensor product double complex \(C \boxtimes D\) is the double complex (in the sense of \(\operatorname {DoubleComplex}_{\mathbb {F}_2}\)) defined as follows:

• Objects: \((C \boxtimes D)_{p,q} = C_p \otimes _{\mathbb {F}_2} D_q\) for all \(p, q \in \mathbb {Z}\).

• Horizontal differential: \(\partial ^h_{p,q} = \partial ^C_p \otimes \operatorname {id}_{D_q} : C_p \otimes D_q \to C_{p-1} \otimes D_q\).

• Vertical differential: \(\partial ^v_{p,q} = \operatorname {id}_{C_p} \otimes \partial ^D_q : C_p \otimes D_q \to C_p \otimes D_{q-1}\).

The double complex conditions are verified as follows. When the complex shape relation \(\operatorname {Rel}(p, p')\) (respectively \(\operatorname {Rel}(q, q')\)) does not hold, the corresponding differential vanishes. The horizontal and vertical differentials each square to zero: \(\partial ^h \circ \partial ^h = 0\) and \(\partial ^v \circ \partial ^v = 0\). Moreover, the horizontal and vertical differentials commute: \(\partial ^h \circ \partial ^v = \partial ^v \circ \partial ^h\), which over \(\mathbb {F}_2\) replaces the usual anticommutativity requirement. This is a consequence of the whisker exchange identity \((\partial ^C \otimes \operatorname {id}) \circ (\operatorname {id} \otimes \partial ^D) = (\operatorname {id} \otimes \partial ^D) \circ (\partial ^C \otimes \operatorname {id})\) in any monoidal category.

The double complex is assembled via HomologicalComplex₂.ofGradedObject using the descending complex shape on \(\mathbb {Z}\) in both directions.

Let \(C\) and \(D\) be chain complexes over \(\mathbb {F}_2\). The tensor product complex \(C \otimes D\) is defined as the total complex of the tensor product double complex:

Concretely, its degree-\(n\) component is

and its differential is \(\partial = \partial ^C \otimes \operatorname {id} + \operatorname {id} \otimes \partial ^D\). Over \(\mathbb {F}_2\), the usual sign factor \((-1)^p\) is trivial since \(-1 = 1\).

Let \(C\) and \(D\) be chain complexes over \(\mathbb {F}_2\), and let \(p, q, n \in \mathbb {Z}\) with \(p + q = n\). The canonical inclusion of the summand \(C_p \otimes D_q\) into \((C \otimes D)_n\) is the map

defined as the total complex inclusion \(\iota _{p,q,n}\) applied to the tensor product double complex \(C \boxtimes D\).

Let \(n_1, m_1, n_2, m_2 \in \mathbb {N}\), and let

be linear maps over \(\mathbb {F}_2\) (parity-check matrices of two classical codes). The hypergraph product code is the chain complex

i.e., the tensor product complex of the parity-check complexes associated to \(\partial ^C\) and \(\partial ^D\). Concretely, the underlying 1-complexes are

and their tensor product double complex has three nontrivial degrees:

• Degree \(2\): \(\mathbb {F}_2^{n_1} \otimes \mathbb {F}_2^{n_2}\),

• Degree \(1\): \((\mathbb {F}_2^{n_1} \otimes \mathbb {F}_2^{m_2}) \oplus (\mathbb {F}_2^{m_1} \otimes \mathbb {F}_2^{n_2})\),

• Degree \(0\): \(\mathbb {F}_2^{m_1} \otimes \mathbb {F}_2^{m_2}\).

The differential from degree \(2\) to degree \(1\) plays the role of \(H_Z^\top \), and the differential from degree \(1\) to degree \(0\) plays the role of \(H_X\) in the CSS code convention. The CSS code structure is read off from the total complex shifted by \(1\) to match the convention \(C_1 \xrightarrow {H_Z^\top } C_0 \xrightarrow {H_X} C_{-1}\).