Association schemes and MacWilliams dualities for generalized Niederreiter–Rosenbloom–Tsfasman posets
Tom 487 / 2012
Streszczenie
Let $\mathbf{P}$ be a poset on the set $[m] \times [n]$, which is given as the disjoint sum of posets on `columns' of $[m]\times[n]$, and let $\check{\mathbf P}$ be the dual poset of $\mathbf{P}$. Then $\mathbf{P}$ is called a generalized Niederreiter–Rosenbloom–Tsfasman poset (gNRTp) if all further posets on columns are weak order posets of the `same type'. Let $G$ (resp. $\check{G})$ be the group of all linear automorphisms of the space $\mathbb F_q^{m \times n}$ preserving the $\mathbf{P}$-weight (resp. $\check{\mathbf{P}}$-weight). We define two partitions of $\mathbb F_q^{m \times n}$, one consisting of `$\mathbf{P}$-orbits' and the other of `$\check{\mathbf P}$-orbits'. If $\mathbf{P}$ is a gNRTp, then they are respectively the orbits under the action of $G$ on $\mathbb F_q^{m \times n}$ and of $\check{G}$ on $\mathbb F_q^{m \times n}$. Then, under the assumption that $\mathbf{P}$ is not an antichain, we show that (1) $\mathbf{P}$ is a gNRTp iff (2) the $\mathbf{P}$-orbit distribution of $C$ uniquely determines the $\check{\mathbf P}$-orbit distribution of $C^\bot$ for every linear code $C$ in $\mathbb F_q^{m \times n}$ iff (3) $G$ acts transitively on each $\mathbf{P}$-orbit iff (4) $\mathbb F_q^{m \times n}$ together with the classes given by `$(u,v)$ belongs to a class iff $u-v$ belongs to a $\mathbf{P}$-orbit' is a symmetric association scheme. Furthermore, a general method of constructing symmetric association schemes is introduced. When $\mathbf{P}$ is a gNRTp, using this, four association schemes are constructed. Some of their parameters are computed and MacWilliams-type identities for linear codes are derived. Also, we report on the recent developments in the theory of poset codes in the Appendix.
