Klemm code[1]
Description
A member of a family of self-dual linear \((4m,4^1 2^{4m-2})_{\mathbb{Z}_4}\) codes. Its generator matrix consists of a sum of the generator matrix of the repetition code and twice the generator matrix of the SPC code [2].
A generator matrix for this code is \begin{align} \begin{pmatrix} 1 & 1 & 1 & \cdots & 1 & 1\\ 0 & 2 & 0 & \cdots & 0 & 2\\ 0 & 0 & 2 & \cdots & 0 & 2\\ 0 & 0 & 0 & \ddots & \vdots & \vdots\\ 0 & 0 & 0 & \cdots & 2 & 2 \end{pmatrix}\,. \tag*{(1)}\end{align}
Cousins
- \([8,4,4]\) extended Hamming code— The binary image of the \(m=1\) Klemm code under the Gray map is the \([8,4,4]\) extended Hamming code [3; Exam. 3.2].
- Repetition code— The generator matrix of the Klemm code consists of a sum of the generator matrix of the repetition code and twice the generator matrix of the SPC code [2].
- \([n,n-1,2]\) Single parity-check (SPC) code— The generator matrix of the Klemm code consists of a sum of the generator matrix of the repetition code and twice the generator matrix of the SPC code [2].
- \(C_{m,r}\) code— The Klemm code at \(m=8\) is the \(C_{m,r=0}\) code with parameters \([32,16,4]_{\mathbb{Z}_4}\) [4].
- Reed-Muller (RM) code— The Klemm code at \(m=4\) is generated by \(\textnormal{RM}(0,4) + 2\textnormal{RM}(3,4)\) [4].
- ZRM code— The Klemm code at \(m=1\) is the ZRM\((1,2)\) code [3; Exam. 4.1].
Member of code lists
Primary Hierarchy
References
- [1]
- M. Klemm, “Selbstduale Codes �ber dem Ring der ganzen Zahlen modulo 4”, Archiv der Mathematik 53, 201 (1989) DOI
- [2]
- S. T. Dougherty, P. Gaborit, M. Harada, A. Munemasa, and P. Sole, “Type IV self-dual codes over rings”, IEEE Transactions on Information Theory 45, 2345 (1999) DOI
- [3]
- Z. X. Wan, Quaternary Codes (WORLD SCIENTIFIC, 1997) DOI
- [4]
- A. Bonnecaze, P. Sole, C. Bachoc, and B. Mourrain, “Type II codes over Z/sub 4/”, IEEE Transactions on Information Theory 43, 969 (1997) DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2025-04-29)
Cite as:
“Klemm code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/klemm, arXiv:2606.11484