NDLI: Codes from incidence matrices of graphs

Content Provider	Springer Nature Link
Author	Dankelmann, P. Key, J. D. Rodrigues, B. G.
Copyright Year	2011
Abstract	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.
Ending Page	393
Page Count	21
Starting Page	373
File Format	PDF
ISSN	09251022
e-ISSN	15737586
Journal	Designs, Codes and Cryptography
Issue Number	1-3
Volume Number	68
Language	English
Publisher	Springer US
Publisher Date	2012-01-03
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Information and Communication, Circuits Codes Data Encryption Linear codes, general Incidence matrices Data Structures, Cryptology and Information Theory Discrete Mathematics in Computer Science Permutation decoding Graphs Coding and Information Theory Block designs Combinatorics Paths and cycles
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics Theoretical Computer Science Computer Science Applications

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