Książka - szczegóły

Let P be a poset on the set [m]×[n], which is given as the disjoint sum of posets on 'columns' of [m]×[n], and let P̌ be the dual poset of P. Then 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. Ǧ) be the group of all linear automorphisms of the space $𝔽_{q}^{m×n}$ preserving the P-weight (resp. P̌-weight). We define two partitions of $𝔽_{q}^{m×n}$, one consisting of 'P-orbits' and the other of 'P̌-orbits'. If P is a gNRTp, then they are respectively the orbits under the action of G on $𝔽_{q}^{m×n}$ and of Ǧ on $𝔽_{q}^{m×n}$. Then, under the assumption that P is not an antichain, we show that (1) P is a gNRTp iff (2) the P-orbit distribution of C uniquely determines the P̌-orbit distribution of $C^{⊥}$ for every linear code C in $𝔽_{q}^{m×n}$ iff (3) G acts transitively on each P-orbit iff (4) $𝔽_{q}^{m×n}$ together with the classes given by '(u,v) belongs to a class iff u-v belongs to a P-orbit' is a symmetric association scheme. Furthermore, a general method of constructing symmetric association schemes is introduced. When 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.