Quaternary RM (QRM) code[1]
Description
A quaternary linear code over \(\mathbb{Z}_4\) whose mod-two reduction 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 mod-two reduction yielding the RM code RM\((r,m)\) [1; Thm. 19].Cousins
- Reed-Muller (RM) code— The mod-two reduction of the QRM\((r,m)\) code is the RM\((r,m)\) code [1; Thm. 19].
- Gray code— The mod-two reduction of the QRM\((r,m)\) code is the RM\((r,m)\) code [1; Thm. 19].
- Self-dual code over \(\mathbb{Z}_q\)— The dual of a QRM\((r,m)\) code is the QRM\((m-r-1,m)\) code [1; Thm. 19].
- Preparata code— The image of the Preparata code under the Gray map is the QRM\((m-2,m)\) code [1; Thm. 19].
- Kerdock code— The image of the Kerdock code under the Gray map is the QRM\((1,m)\) code, an extended cyclic code over \(\mathbb{Z}_4\) [1; Thm. 19] (see also Ref. [2]).
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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]
- A. A. NECHAEV, “Kerdock code in a cyclic form”, Discrete Mathematics and Applications 1, (1991) 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