Here is a list of codes related to the Gray code in some way.
| Code | Relation |
|---|---|
| Delsarte-Goethals (DG) code | DG codes can be seen, via the Gray map, as extended linear cyclic codes over \(\mathbb{Z}_4\) [1]. |
| Gray code | |
| Hergert code | Hergert codes can be seen, via the Gray map, as linear codes over \(\mathbb{Z}_4\) [1,2]. |
| Julin-Golay code | Julin codes can be obtained from simple nonlinear codes over \(\mathbb{Z}_4\) using the Gray map [3]. |
| Kerdock code | The binary Kerdock code is the Gray-map image of the quaternary code QRM\((1,m)\), an extended cyclic code over \(\mathbb{Z}_4\) [1; Thm. 19] (see also Ref. [4]). |
| Linear code over \(\mathbb{Z}_4\) | A linear code \(C\) over \(\mathbb{Z}_4\) can be mapped, via the Gray map, to a binary code. The binary code is linear if and only if doubling the component-wise product of any two codewords in \(C\) yields another codeword in \(C\) [5; Thm. 12.2.3]. More specifically, a linear quaternary code over \(\mathbb{Z}_4\) of length \(n\), type \(4^{k_1}2^{k_2}\), and minimum Lee weight \(d\) maps under the Gray map to a binary code of length \(2n\), cardinality \(2^{2k_1+k_2}\), and minimum Hamming weight \(d\) [6; Sec. 6.3]. |
| Phase-shift keying (PSK) code | 1D Gray codes are often concatenated with PSKs so that the Hamming distance between the bitstrings encoded into the points is a discretized version of the Euclidean distance between the points. |
| Preparata code | The binary Preparata code is the Gray-map image of the quaternary code QRM\((m-2,m)\) [1; Thm. 19]. |
| Quadrature-amplitude modulation (QAM) code | 2D Gray codes are often concatenated with \(n=1\) lattice-based QAM codes so that the Hamming distance between the bitstrings encoded into the points is a discretized version of the Euclidean distance between the points. |
| Qubit code | Gray codes are useful for optimizing qubit unitary circuits [7] and for encoding qudits in multiple qubits [8]. |
| Rank-modulation code | The rank-modulation Gray code is an extension of the original binary Gray code to a code on the permutation group [9]. |
| Self-dual code over \(\mathbb{Z}_4\) | Under the Gray map, any self-dual code over \(\mathbb{Z}_4\) maps to a formally self-dual binary code [10]. |
| ZRM code | The image of the ZRM\((r,m-1)\) code under the Gray map is the RM\((r,m)\) code [1; Thm. 7]. |
| \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code | A construction using the \(\mathbb{Z}_4\) versions of the Goethals and Preparata codes and the Gray map yields qubit code families with similar parameters as the \(((2^m,2^{2^m−5m+1},8))\) Goethals-Preparata code [11]. |
| \((10,40,4)\) Best code | Codewords of the Best code can be obtained by applying the Gray map to the pentacode [3; Sec. 2]. |
References
- [1]
- A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Solé, “The Z_4-Linearity of Kerdock, Preparata, Goethals and Related Codes”, (2002) arXiv:math/0207208
- [2]
- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
- [3]
- J. H. Conway and N. J. A. Sloane, “Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others”, Designs, Codes and Cryptography 4, 31 (1994) DOI
- [4]
- A. A. NECHAEV, “Kerdock code in a cyclic form”, Discrete Mathematics and Applications 1, (1991) DOI
- [5]
- W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, 2003) DOI
- [6]
- S. T. Dougherty, “Codes over rings.” Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [7]
- J. J. Vartiainen, M. Möttönen, and M. M. Salomaa, “Efficient Decomposition of Quantum Gates”, Physical Review Letters 92, (2004) arXiv:quant-ph/0312218 DOI
- [8]
- N. P. D. Sawaya, T. Menke, T. H. Kyaw, S. Johri, A. Aspuru-Guzik, and G. G. Guerreschi, “Resource-efficient digital quantum simulation of d-level systems for photonic, vibrational, and spin-s Hamiltonians”, npj Quantum Information 6, (2020) arXiv:1909.12847 DOI
- [9]
- Anxiao Jiang, R. Mateescu, M. Schwartz, and J. Bruck, “Rank Modulation for Flash Memories”, IEEE Transactions on Information Theory 55, 2659 (2009) DOI
- [10]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
- [11]
- S. Ling and P. Sole. 2008. Nonadditive quantum codes from Z4-codes. hal.archives-ouvertes.fr/hal-00338309/fr