Designs, Codes and Cryptography

, Volume 68, Issue 1–3, pp 373–393 | Cite as

Codes from incidence matrices of graphs



We examine the p-ary codes, for any prime p, from the row span over \({\mathbb {F}_p}\) of |V| × |E| incidence matrices of connected graphs Γ = (V, E), showing that certain properties of the codes can be directly derived from the parameters and properties of the graphs. Using the edge-connectivity of Γ (defined as the minimum number of edges whose removal renders Γ disconnected) we show that, subject to various conditions, the codes from such matrices for a wide range of classes of connected graphs have the property of having dimension |V| or |V| − 1, minimum weight the minimum degree δ(Γ), and the minimum words the scalar multiples of the rows of the incidence matrix of this weight. We also show that, in the k-regular case, there is a gap in the weight enumerator between k and 2k − 2 of the binary code, and also for the p-ary code, for any prime p, if Γ is bipartite. We examine also the implications for the binary codes from adjacency matrices of line graphs. Finally we show that the codes of many of these classes of graphs can be used for permutation decoding for full error correction with any information set.


Incidence matrices Graphs Codes Permutation decoding 

Mathematics Subject Classification (2010)

05B05 05C38 94B05 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of JohannesburgAuckland ParkSouth Africa
  2. 2.School of Mathematical SciencesUniversity of KwaZulu-NatalDurbanSouth Africa

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