Higman-Sims graph-adjacency code

Higman-Sims graph-adjacency code[1,2]

Description

A graph-based code whose generator matrix is constructed using the adjacency matrix of the Higman-Sims graph. This family includes a \([100,22,32]\) code whose dual is an optimal \([100,78,8]\) code [1; Table VI].

Cousins

Combinatorial design— Codewords of weight 36 of the Higman-Sims graph-adjacency code form a \(2\)-\((100,36,525)\) design [1; Remark 1.7]
\(\Lambda_{24}\) Leech lattice— The Higman-Sims graph occurs in the Leech lattice [3].
Two-point homogeneous-space code— The Higman-Sims graph is distance-transitive, hence it is a finite two-point homogeneous space [4].

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC

Graph-adjacency code

Parents

Graph-adjacency code

Higman-Sims graph-adjacency code

References

[1]: V. D. Tonchev, “Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs”, IEEE Transactions on Information Theory 43, 1021 (1997) DOI
[2]: V. D. Tonchev, “Error-correcting codes from graphs”, Discrete Mathematics 257, 549 (2002) DOI
[3]: J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups (Springer New York, 1999) DOI
[4]: A. E. Brouwer, A. M. Cohen, and A. Neumaier, Distance-Regular Graphs (Springer Berlin Heidelberg, 1989) DOI

Page edit log

Andrey Boris Khesin (2025-01-31) — most recent
Victor V. Albert (2025-01-31)
Victor V. Albert (2024-03-21)

Cite as:

“Higman-Sims graph-adjacency code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/higman-sims_graph

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/graph/adjacency/higman-sims_graph.yml.