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Binary codes and partial permutation decoding sets from biadjacency matrices of bipartite graphs Γ(2k, k, k + 1, 1)
Quaestiones Mathematicae ( IF 0.6 ) Pub Date : 2019-03-17 , DOI: 10.2989/16073606.2019.1580227
W. Fish 1 , N.B. Mumba 2 , E. Mwambene 1 , B.G. Rodrigues 3
Affiliation  

Abstract For a set Ω = {1, 2, . . . , n} where n = 2k ≥ 6, let Ω{k} denote the set of all subsets of Ω of size k. We examine the binary codes from the row span of biadjacency matrices of bipartite graphs with bipartition (Ω{k}, Ω{k+1}) and two vertices as k-subsets and (k+1)-subsets of Ω being adjacent if they have one element in common. We show that S2k is contained in the automorphism group of the graphs and the codes, respectively. In addition, we determine the duals of the codes, and by identifying suitable information sets, we construct 2-PD sets for the dual codes.

中文翻译:

二分图 Γ(2k, k, k + 1, 1) 的双邻接矩阵的二进制码和部分置换解码集

摘要 对于集合 Ω = {1, 2, . . . , n} 其中 n = 2k ≥ 6,令 Ω{k} 表示大小为 k 的 Ω 的所有子集的集合。我们检查来自二分图的双邻接矩阵的行跨度的二进制代码,二分图 (Ω{k}, Ω{k+1}) 和两个顶点作为 k 子集和 Ω 的 (k+1) 子集相邻,如果它们有一个共同点。我们证明 S2k 分别包含在图和代码的自同构群中。此外,我们确定了代码的对偶,并通过识别合适的信息集,我们为对偶代码构建了 2-PD 集。
更新日期:2019-03-17
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