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3 Open Mathematics

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Multiples of left loops and vertex-transitive graphs

100%

Mwambene E.

nr 2

245-250

Via representation of vertex-transitive graphs on groupoids, we show that left loops with units are factors of groups, i.e., left loops are transversals of left cosets on which it is possible to define a binary operation which allows left cancellation.

Binary codes and partial permutation decoding sets from the odd graphs

51%

Fish W. , Fray R. , Mwambene E.

nr 9

1362-1371

For k ≥ 1, the odd graph denoted by O(k), is the graph with the vertex-set Ω{k}, the set of all k-subsets of Ω = {1, 2, …, 2k +1}, and any two of its vertices u and v constitute an edge [u, v] if and only if u ∩ v = /0. In this paper the binary code generated by the adjacency matrix of O(k) is studied. The automorphism group of the code is determined, and by identifying a suitable information set, a 2-PD-set of the order of k 4 is determined. Lastly, the relationship between the dual code from O(k) and the code from its graph-theoretical complement $\overline {O(k)} $, is investigated.

Codes and designs from triangular graphs and their line graphs

51%

Fish W. , Kumwenda K. , Mwambene E.

nr 6

1411-1423

For any prime p, we consider p-ary linear codes obtained from the span over $\mathbb{F}_p $ p of rows of incidence matrices of triangular graphs, differences of the rows and adjacency matrices of line graphs of triangular graphs. We determine parameters of the codes, minimum words and automorphism groups. We also show that the codes can be used for full permutation decoding.