Abstract
For \(k \ge 1\), \(n \ge 2k\), the Johnson graph denoted by \(J(n,k),\) is the graph with vertex-set the set of all \(k\)-subsets of \(\Omega = \{1, 2, \ldots , n\}\), and any two vertices \(u\) and \(v\) are adjacent if and only if \(|u \cap v| = k-1\). In this paper the binary codes and their duals generated by an adjacency matrix of \(J(n,k)\) are described. The automorphism groups of the codes are determined, and by identifying suitable information sets, 3-PD-sets are determined for the code when \(k\) is even.
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Acknowledgments
The author is indebted to Professor J.D. Key from the Department of Mathematics and Applied Mathematics at the University of the Western Cape for her advice and encouragement.
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Fish, W. Binary Codes and Partial Permutation Decoding Sets from the Johnson Graphs. Graphs and Combinatorics 31, 1381–1396 (2015). https://doi.org/10.1007/s00373-014-1485-2
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DOI: https://doi.org/10.1007/s00373-014-1485-2