Graph-adjacency code

Graph-adjacency code[1,2]

Description

Binary linear code whose generator matrix forms the adjacency matrix of a strongly regular graph. Given an adjacency matrix \(A\), the generator matrix is either \(G=A\) or \(G=(I|A)\), where \(I\) is the identity matrix.

Codes based on strongly regular graphs are sometimes optimal or nearly optimal for their length and size [1].

Cousins

Cycle code— Graph-adjacency (cycle) codes' generator (parity-check) matrices are defined using adjacency (incidence) matrices of graphs.
Hermitian qubit code— Bounds on self-dual \([[n,0,d]]\) Hermitian codes based on graphs have been derived [2].

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary ECC

Parents

Linear binary code

Graph-adjacency code

Children

Higman-Sims graph-adjacency code

Hoffman-Singleton 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

Page edit log

Victor V. Albert (2024-01-06) — most recent

Cite as:

“Graph-adjacency code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/graph

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