Description
Spherical \((n,2^n,4/n)\) code whose codewords are vertices of an \(n\)-cube, i.e., all permutations and negations of the vector \((1,1,\cdots,1)\), up to normalization.
Parents
- Polytope code — Hypercube codewords in 2 (3, 4, \(n\)) dimensions form the vertices of a triangle (square, cube, \(n\)-cube).
- Lattice-shell code — Hypercube codewords form the minimal lattice shell code of the \(\mathbb{Z}^n\) hypercubic lattice when the lattice is shifted such that the center of a hypercube is at the origin.
- Spherical design — Hypercube codes form spherical 3-designs. The weighted union of the vertices of a hypercube and a cross polytope form a weighted spherical 5-design in dimensions \(\geq 3\) [1; Exam. 2.6].
- Polyphase code
Cousins
- Biorthogonal spherical code — Orthoplexes and hypercubes are dual to each other.
- Dual polytope code — Orthoplexes and hypercubes are dual to each other.
- \(\mathbb{Z}^n\) hypercubic lattice — Hypercube codewords form the minimal lattice shell code of the \(\mathbb{Z}^n\) hypercubic lattice when the lattice is shifted such that the center of a hypercube is at the origin.
- Binary antipodal code — Binary antipodal codes are subcodes of a hypercube code since the hypercube code corresponds to the Hamming \(n\)-cube (a.k.a. Boolean hypercube) embedded into the unit \(n\)-sphere.
- Binary code — Binary strings are elements of the Hamming \(n\)-cube (a.k.a. Boolean hypercube).
- 24-cell code — The vertices of a 24-cell are a union of the vertices of a tesseract and a 16-cell [1; Exam. 2.6].
- \([[2^D,D,2]]\) hypercube quantum code — \([[2^D,D,2]]\) hypercube quantum code qubits are placed on vertices of a \(D\)-cube.
- \([[m 2^m / (m+1), 2^m / (m+1)]]\) Khesin-Lu-Shor code — The encoder-respecting form of the \([[m 2^m / (m+1), 2^m / (m+1)]]\) Khesin-Lu-Shor code is the graph of a hypercube in \(m = 2^r - 1\) dimensions, and input nodes in the graph are codewords of the \([2^r-1,2^r-r-1,3]\) Hamming code [2].
- \([[16,6,4]]\) Tesseract color code — Stabilizer generators of both types of the tesseract color code are supported on each cube of a tesseract [3,4].
- Hemicubic code
References
- [1]
- S. Borodachov, P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Energy bounds for weighted spherical codes and designs via linear programming”, (2024) arXiv:2403.07457
- [2]
- J. Z. Lu, A. B. Khesin, and P. W. Shor, “Universal graph representation of stabilizer codes”, (2024) arXiv:2411.14448
- [3]
- N. Delfosse and B. W. Reichardt, “Short Shor-style syndrome sequences”, (2020) arXiv:2008.05051
- [4]
- P. Prabhu and B. W. Reichardt, “Distance-four quantum codes with combined postselection and error correction”, Physical Review A 110, (2024) arXiv:2112.03785 DOI
Page edit log
- Victor V. Albert (2024-03-21) — most recent
Cite as:
“Hypercube code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hypercube