Alternative Names: Galois-qudit quantum hypergraph (QHG) code, Galois-qudit Tillich-Zemor product code.
Description
A member of a family of Galois-qudit CSS codes whose stabilizer generator matrix is obtained from a hypergraph product of two classical linear \(q\)-ary codes.Cousins
- Linear \(q\)-ary code— Galois-qudit HGP codes are constructed out of two classical linear \(q\)-ary codes.
- Two-block group-algebra (2BGA) codes— A 2BGA code \(LP(a,b)\) is constructible as a hypergraph-product code when the support subgroups generated by \(a\) and \(b\) are disjoint. In that case, the commuting matrices simultaneously acquire hypergraph-product Kronecker-product form, and the code can be obtained from a pair of classical group-algebra codes [1; Statements 8 and 12].
- Abelian two-block group-algebra code— An Abelian 2BGA code whose elements \(a\) and \(b\) are supported on subgroups intersecting trivially is a square-matrix hypergraph-product code constructed from a pair of classical group-algebra codes [1; Statements 8 and 12]. Elements given by polynomials in disjoint sets of variables satisfy this condition. See the cyclic HGP code entry for the case of two cyclic codes.
Primary Hierarchy
Balanced product (BP) codeGeneralized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
Lifted-product codes for trivial lift are Galois-qudit hypergraph-product codes.
Galois-qudit HGP code
Children
Hypergraph product codes are Galois-qudit hypergraph-product codes for qudit dimension \(q=2\).
References
- [1]
- H.-K. Lin and L. P. Pryadko, “Quantum two-block group algebra codes”, (2023) arXiv:2306.16400
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-10-22)
Cite as:
“Galois-qudit HGP code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/galois_hypergraph_product, arXiv:2606.11484