Description
Binary code that detects any single bit flip on four bits by appending one parity bit to three message bits.Cousins
- \([[4,2,2]]\) Four-qubit code— The \([[4,2,2]]\) code is constructed from the \([4,3,2]\) SPC code via the CSS construction.
- \([[4,1,2]]\) Leung-Nielsen-Chuang-Yamamoto (LNCY) code— The \(Z\)-type stabilizer of the \([[4,1,2]]\) LNCY code is the parity check of the \([4,3,2]\) SPC code.
- \(D_4\) hyper-diamond lattice— The \(D_4\) lattice is constructed out of the \([4,3,2]\) SPC code via Construction A [1; pg. 138].
Member of code lists
- Algebraic-geometry codes
- Binary linear codes
- Classical codes
- Cyclic codes
- Evaluation codes
- Locally correctable codes and friends
- Locally recoverable codes
- MDS codes and generalizations
- Orthogonal arrays and friends
- Projective codes
- Reed-Solomon codes and friends
- Small-distance classical codes and friends
- Universally optimal codes
Primary Hierarchy
\([n,n-1,2]\) Single parity-check (SPC) codeGRS Evaluation MDS Divisible Linear \(q\)-ary AG Nearly perfect OA Sharp configuration Universally optimal LRC Distributed-storage Small-distance block Cyclic Constacyclic Skew-cyclic ECC \(t\)-design
Parents
\([2^m,m+1,2^{m-1}]\) First-order RM codeGRM Evaluation AG Divisible Projective geometry Linear \(q\)-ary Linear code over \(\mathbb{Z}_q\) LCC LRC Distributed-storage ECC
The \([4,3,2]\) SPC code is the RM\((1,2)\) code [2; Ch. 13, Prob. 5].
\([4,3,2]\) single parity-check (SPC) code
References
- [1]
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
- [2]
- F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (Elsevier, 1977)
Page edit log
- Victor V. Albert (2026-10-13) — most recent
Cite as:
“\([4,3,2]\) single parity-check (SPC) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/spc_4_3_2, arXiv:2606.11484