Quaternary RM (QRM) code[1]
Alternative names: \(\mathbb{Z}_4\) RM code.
Description
A quaternary linear code over \(\mathbb{Z}_4\) that is a quaternary version of the RM code in that its binary image under the Gray map is an RM code. This code subsumes the quaternary images of the Kerdock and Preparata codes under the Gray map . The code is usually noted as QRM\((r,m)\), with its image under the Gray map yielding the RM code RM\((r,m)\) [1; Thm. 19].Cousins
- Reed-Muller (RM) code— The image of the QRM\((r,m)\) code under the Gray map is the RM\((r,m)\) code [1; Thm. 19][2; Sec. 6.3]. Unions of certain RM codes yield self-dual quaternary codes over \(\mathbb{Z}_4\) that then give rise to certain BW lattices [3,4].
- Gray code— The image of the QRM\((r,m)\) code under the Gray map is the RM\((r,m)\) code [1; Thm. 19][2; Sec. 6.3].
- Preparata code— The image of the Preparata code under the Gray map yields the QRM\((m-2,m)\) code [1; Thm. 19].
- Kerdock code— The image of the Kerdock code under the Gray map yields the QRM\((1,m)\) code [1; Thm. 19].
Member of code lists
Primary Hierarchy
References
- [1]
- A. R. Hammons Jr., 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]
- S. T. Dougherty, "Codes over rings." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [3]
- P. Sole, "Generalized theta functions for lattice vector quantization", in Coding and Quantization, DIMACS Series in Dr,crete Mathenulies and Theoretical Computer Science, vol. 14. Providence, RH: American Math. Soc., 1993, pp. 27-32.
- [4]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
Page edit log
- Victor V. Albert (2024-08-15) — most recent
Cite as:
“Quaternary RM (QRM) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/quaternary_reed_muller