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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].

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
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Zoo Code ID: zrm

Cite as:
“ZRM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/zrm
BibTeX:
@incollection{eczoo_zrm, title={ZRM code}, booktitle={The Error Correction Zoo}, year={2025}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/zrm} }
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https://errorcorrectionzoo.org/c/zrm

Cite as:

“ZRM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/zrm

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/rings/over_zq/over_z4/linear_over_z4/rm/zrm.yml.