Fourier-Reflexive Partitions Induced by Poset Metric
Yang Xu, Haibin Kan, Guangyue Han · IEEE Transactions on Information Theory · 2021
Let$\mathbf {H}$be the cartesian product of a family of finite abelian groups indexed by a finite set$\Omega $. A given poset (i.e., partially ordered set)$\mathbf {P}=(\Omega,\preccurlyeq _{\mathbf {P}})$gives rise to a poset metric on$\mathbf {H}$, which further leads to a partition$\mathcal {Q}(\mathbf {H},\mathbf {P})$of$\mathbf {H}$. We prove that if$\mathcal {Q}(\mathbf {H},\mathbf {P})$is Fourier-reflexive, then its dual partition$\Lambda $coincides with the partition of$\hat {\mathbf {H}}$induced by$\mathbf {\overline {P}}$, the dual poset of$\mathbf {P}$, and moreover,$\mathbf {P}$is necessarily hierarchical. This result establishes a conjecture proposed by Gluesing-Luerssen in Gluesing-Luerssen, 2015. We also show that with some other assumptions,$\Lambda $is finer than the partition of$\hat {\mathbf {H}}$induced by$\mathbf {\overline {P}}$. In addition, we give some necessary and sufficient conditions for$\mathbf {P}$to be hierarchical, and for the case that$\mathbf {P}$is hierarchical, we give an explicit criterion for determining whether two codewords in$\hat {\mathbf {H}}$belong to the same block of$\Lambda $. We prove these results by relating the involved partitions with certain family of polynomials, a generalized version of which is also proposed and studied to generalize the aforementioned results.