Description
The \((22,275,1/6)\) or \((23,552,1/5)\) code associated with the McLaughlin graph and the Leech lattice. See Ref. [3] for explicit constructions of and relations between both codes.Cousins
- Spherical design— Both McLaughlin spherical codes are sharp configurations [3,4]. The \((22,275,1/6)\) code is a unique and tight spherical 4-design, while the \((23,552,1/5)\) code is a unique and tight spherical 5-design; see Ref. [4; Appx. A].
- \([23, 12, 7]\) Golay code— The McLaughlin spherical code can be constructed from length-23 Golay codewords [3].
- Real projective space code— The \((23,552,1/5)\) McLaughlin spherical code yields a set of \(276\) equiangular lines in 23 dimensions [3,4].
- \(\Lambda_{24}\) Leech lattice-shell code— The \((23,552,1/5)\) McLaughlin code can be derived from a shell of the Leech lattice [4,5].
Member of code lists
Primary Hierarchy
Spherical sharp configurationSpherical design Sharp configuration Universally optimal ECC \(t\)-design
Parents
Both McLaughlin spherical codes are sharp configurations [3,4]. The \((22,275,1/6)\) code is a unique and tight spherical 4-design, while the \((23,552,1/5)\) code is a unique and tight spherical 5-design; see Ref. [4; Appx. A].
McLaughlin spherical code
References
- [1]
- J. McLaughlin, “A simple group of order 898,128,000”, Theory of finite groups. Benjamin New York, 1969. 109-111
- [2]
- J. M. GOETHALS and J. J. SEIDEL, “THE REGULAR TWO-GRAPH ON 276 VERTICES”, Geometry and Combinatorics 177 (1991) DOI
- [3]
- P. Boyvalenkov, P. Dragnev, D. Hardin, E. Saff, and M. Stoyanova, “Universal minima of discrete potentials for sharp spherical codes”, (2023) arXiv:2211.00092
- [4]
- H. Cohn and A. Kumar, “Universally optimal distribution of points on spheres”, Journal of the American Mathematical Society 20, 99 (2006) arXiv:math/0607446 DOI
- [5]
- M. Dutour Sikirić, A. Schürmann, and F. Vallentin, “The Contact Polytope of the Leech Lattice”, Discrete & Computational Geometry 44, 904 (2010) arXiv:0906.1427 DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2022-11-28)
Cite as:
“McLaughlin spherical code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/mclaughlin, arXiv:2606.11484