Description
A \([50,21,12]\) cycle code whose parity-check matrix is the incidence matrix of the Hoffman-Singleton graph [3]. Its dual is a \([50,29,8]\) code [1; Table II].
The Hoffman-Singleton graph-adjacency code \([50,22,7]\) contains this code as a subcode of codimension 1 [1]. The dual \([50,29,8]\) code admits majority decoding correcting up to two errors via Rudolph’s algorithm applied to the \(2\text{-}(50,14,13)\) design derived from the Hoffman-Singleton graph [1; Cor. 1.3].
Cousins
- Hoffman-Singleton graph-adjacency code— The Hoffman-Singleton cycle code and graph-adjacency code are both derived from the Hoffman-Singleton graph.
- Combinatorial design— The incidence matrix of the Hoffman-Singleton graph can be converted into a \(2\)-\((50,14,13)\) design [1; Prop. 1.1].
Member of code lists
Primary Hierarchy
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]
- A. J. Hoffman and R. R. Singleton, “On Moore Graphs with Diameters 2 and 3”, IBM Journal of Research and Development 4, 497 (1960) DOI
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Victor V. Albert (2024-03-21)
Cite as:
“Hoffman-Singleton cycle code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/hoffman-singleton, arXiv:2606.11484