\(C_{m,r}\) code[1]
Description
A member of a family of Type IV self-dual quaternary linear codes over \(\mathbb{Z}_4\) generated by \(\textnormal{RM}(r,m) + 2\textnormal{RM}(m-r-1,m)\) for \(3r \leq m-1\) [2].Cousins
- Reed-Muller (RM) code— The \(C_{m,r}\) code is generated by \(\textnormal{RM}(r,m) + 2\textnormal{RM}(m-r-1,m)\) for \(3r \leq m-1\) [2].
- Barnes-Wall (BW) lattice— \(C_{m,r=1}\) codes give rise to certain BW lattices [3,4].
- \(BW_{32}\) Barnes-Wall lattice— The \(C_{m=5,r=1}\) code gives rise to the \(B_{32}\) Barnes-Wall lattice via Construction \(A_4\) [3,4].
- Klemm code— The Klemm code at \(m=8\) is the \(C_{m,r=0}\) code with parameters \([32,16,4]_{\mathbb{Z}_4}\) [1].
Member of code lists
Primary Hierarchy
References
- [1]
- 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
- [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]
- P. Sole, "Generalized theta functions for lattice vector quantization", in Coding and Quantization, DIMACS Series in Dr,crete Mathenulies and Theoretical Computer Science, vol. 14. Providence, RH: American Math. Soc., 1993, pp. 27-32.
- [4]
- A. Bonnecaze, P. Sole, and A. R. Calderbank, “Quaternary quadratic residue codes and unimodular lattices”, IEEE Transactions on Information Theory 41, 366 (1995) DOI
Page edit log
- Victor V. Albert (2025-04-29) — most recent
Cite as:
“\(C_{m,r}\) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/cmr