Relations among certain classes of incidence loops, codes and chain structures. (Q1411455)

When investigating loops with the property that the \({\mathcal Z}(a):=\{x\in L\mid a+x=x+a\}\) are commutative subgroups and form a \(0\)-bundle \(\mathcal F\) (i.e. \(| A|\geq 2\), \(A\cap B=\{0\}\;\forall A,B\in{\mathcal F}\), \(A\neq B\) and \(\bigcup_{X\in{\mathcal F}}X=L\)) the first case, which comes into mind, is if \({\mathcal Z}(a)\cong\mathbb{Z}_2\), i.e. \(a+a=0\;\forall a\in L\). This case is considered here and connections with certain codes are displayed.