Quaternary RM (QRM) code[1]
Description
A quaternary linear code over \(\mathbb{Z}_4\) whose binary 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].Decoding
QRM codes that are the images of Preparata codes under the Gray map can be decoded using a syndrome-calculation-based algorithm to correct all error patterns of Lee weight at most 2 and detect all (or, for some constructions, a subset of) error patterns of Lee weight 3 or 4 [1,2].Cousins
- Reed-Muller (RM) code— The mod-two reduction of the QRM\((r,m)\) code is the RM\((r,m)\) code [1; Thm. 19].
- Dual code over \(\mathbb{Z}_4\)— 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. [3]).
Member of code lists
Primary Hierarchy
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]
- A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, “The Z/sub 4/-linearity of Kerdock, Preparata, Goethals, and related codes”, IEEE Transactions on Information Theory 40, 301 (1994) DOI
- [3]
- 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