ZRM code[1]
Description
A quaternary linear code over \(\mathbb{Z}_4\) that reduces to the RM code under the Gray map. The code is usually denoted as ZRM\((r,m-1)\), with its image under the Gray map being the RM code RM\((r,m)\) [1; Thm. 7]. The code is generated by \(\textnormal{RM}(r-1,m-1) + 2\textnormal{RM}(r,m-1)\) [1; Thm. 7].Cousins
- Reed-Muller (RM) code— The ZRM code is generated by \(\textnormal{RM}(r-1,m-1) + 2\textnormal{RM}(r,m-1)\) [1; Thm. 7]. The image of the ZRM\((r,m-1)\) code under the Gray map is the RM\((r,m)\) code [1; Thm. 7].
- Gray code— The image of the ZRM\((r,m-1)\) code under the Gray map is the RM\((r,m)\) code [1; Thm. 7].
- Combinatorial design— The weight-four codewords of the binary image of the dual of ZRM\((1,m)\) form a Steiner system that is identical to that formed by the weight-four codewords of an extended Hamming code [1].
- \([2^r,2^r-r-1,4]\) Extended Hamming code— The weight-four codewords of the binary image of the dual of ZRM\((1,m)\) form a Steiner system that is identical to that formed by the weight-four codewords of an extended Hamming code [1].
- Preparata code— Each Preparata code is contained in a corresponding dual of ZRM\((1,m)\) [1].
- Kerdock code— Each Kerdock code is contained in a corresponding ZRM\((2,m)\) code [1].
- Klemm code— The Klemm code at \(m=1\) is the ZRM\((1,2)\) code [2; Exam. 4.1].
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]
- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
Page edit log
- Victor V. Albert (2025-04-29) — most recent
Cite as:
“ZRM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/zrm