Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets
Dae San Kim ; Hyun Kwang Kim
GDML_Books, (2012), p.

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 qm×n preserving the P-weight (resp. P̌-weight). We define two partitions of qm×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 qm×n and of Ǧ on qm×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 qm×n iff (3) G acts transitively on each P-orbit iff (4) qm×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.

EUDML-ID : urn:eudml:doc:285957
@book{bwmeta1.element.bwnjournal-rm-doi-10_4064-dm487-0-1,
     author = {Dae San Kim and Hyun Kwang Kim},
     title = {Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets},
     series = {GDML\_Books},
     year = {2012},
     zbl = {1269.05119},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm487-0-1}
}
Dae San Kim; Hyun Kwang Kim. Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets. GDML_Books (2012),  http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-rm-doi-10_4064-dm487-0-1/