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].

Parent

Children

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].

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
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Zoo Code ID: graph

Cite as:
“Graph-adjacency code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/graph
BibTeX:
@incollection{eczoo_graph, title={Graph-adjacency code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/graph} }
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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.