Permutation decoding was introduced by MacWilliams [Mac64] in the early 60’s. It can be used when... more Permutation decoding was introduced by MacWilliams [Mac64] in the early 60’s. It can be used when a linear code has a sufficiently large automorphism group to ensure the existence of a set of automorphisms, called a PD-set, that has some specifed properties. This series of talks will describe the method and some recent developments in finding PD-sets for codes defined through the row-span over finite fields of incidence matrices of classes of designs or graphs, and adjacency matrices of classes of regular graphs. These codes have many properties that can be deduced from the combinatorial properties of the designs or graphs, and often have a great deal of symmetry and large automorphism groups. J.D.Key (keyj@clemson.edu) Permutation decoding ASI Croatia June 2010 2 / 81
Abstract We examine the p-ary codes, for any prime p, from the row span over F _p of| V|×| E| inc... more Abstract We examine the p-ary codes, for any prime p, from the row span over 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.
... Hamming graphs $H^k(n,2)$ for $k \geq 2$. doi:10.3934/amc.2011.5.373 Full text: (518.6K) Jenn... more ... Hamming graphs $H^k(n,2)$ for $k \geq 2$. doi:10.3934/amc.2011.5.373 Full text: (518.6K) Jennifer D. Key - Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellville, South Africa (email). ...
We show that permutation decoding can be used, and give explicit PD-sets in the symmetric group, ... more We show that permutation decoding can be used, and give explicit PD-sets in the symmetric group, for some of the binary codes obtained from the adjacency matrices of the graphs on (n 3) vertices, for n≥ 7, with adjacency defined by the vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively.
Permutation decoding was introduced by MacWilliams [Mac64] in the early 60’s. It can be used when... more Permutation decoding was introduced by MacWilliams [Mac64] in the early 60’s. It can be used when a linear code has a sufficiently large automorphism group to ensure the existence of a set of automorphisms, called a PD-set, that has some specifed properties. This series of talks will describe the method and some recent developments in finding PD-sets for codes defined through the row-span over finite fields of incidence matrices of classes of designs or graphs, and adjacency matrices of classes of regular graphs. These codes have many properties that can be deduced from the combinatorial properties of the designs or graphs, and often have a great deal of symmetry and large automorphism groups. J.D.Key (keyj@clemson.edu) Permutation decoding ASI Croatia June 2010 2 / 81
Abstract We examine the p-ary codes, for any prime p, from the row span over F _p of| V|×| E| inc... more Abstract We examine the p-ary codes, for any prime p, from the row span over 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.
... Hamming graphs $H^k(n,2)$ for $k \geq 2$. doi:10.3934/amc.2011.5.373 Full text: (518.6K) Jenn... more ... Hamming graphs $H^k(n,2)$ for $k \geq 2$. doi:10.3934/amc.2011.5.373 Full text: (518.6K) Jennifer D. Key - Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellville, South Africa (email). ...
We show that permutation decoding can be used, and give explicit PD-sets in the symmetric group, ... more We show that permutation decoding can be used, and give explicit PD-sets in the symmetric group, for some of the binary codes obtained from the adjacency matrices of the graphs on (n 3) vertices, for n≥ 7, with adjacency defined by the vertices as 3-sets being adjacent if they have zero, one or two elements in common, respectively.
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