Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets

Abstract

Let P be a poset on the set [m] x [n], which is given as the disjoint sum of posets on 'columns' of [m] x [n], and let P 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. (sic)) be the group of all linear automorphisms of the space F-q(m x n) preserving the P-weight (resp. (sic)-weight). We define two partitions of F-q(m x n), one consisting of 'P-orbits' and the other of '(sic)-orbits'. If P is a gNRTp, then they are respectively the orbits under the action of G on F-q(m x n) and of (sic) on F-q(m x n). Then, under the assumption that P is not an antichain, we show that (1) P is a gNRTp if (2) the P-orbit distribution of C uniquely determines the (sic)-orbit distribution of C-perpendicular to for every linear code C in F-q(m x n) iff (3) G acts transitively on each P-orbit iff (4) F-q(m x 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.