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

Dae San Kim, and Hyun Kwang Kim. Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets. 2012. <http://eudml.org/doc/285957>.

@book{DaeSanKim2012,
abstract = {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 $_\{q\}^\{m×n\}$ preserving the P-weight (resp. P̌-weight). We define two partitions of $_\{q\}^\{m×n\}$, 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 $_\{q\}^\{m×n\}$ and of Ǧ on $_\{q\}^\{m×n\}$. 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 $_\{q\}^\{m×n\}$ iff (3) G acts transitively on each P-orbit iff (4) $_\{q\}^\{m×n\}$ 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.},
author = {Dae San Kim, Hyun Kwang Kim},
keywords = {association schemes; MacWilliams-type identity; weak dual orbit pair; generalized Niederreiter-Rosenbloom-Tsfasman poset},
language = {eng},
title = {Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets},
url = {http://eudml.org/doc/285957},
year = {2012},
}

TY - BOOK
AU - Dae San Kim
AU - Hyun Kwang Kim
TI - Association schemes and MacWilliams dualities for generalized Niederreiter-Rosenbloom-Tsfasman posets
PY - 2012
AB - 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 $_{q}^{m×n}$ preserving the P-weight (resp. P̌-weight). We define two partitions of $_{q}^{m×n}$, 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 $_{q}^{m×n}$ and of Ǧ on $_{q}^{m×n}$. 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 $_{q}^{m×n}$ iff (3) G acts transitively on each P-orbit iff (4) $_{q}^{m×n}$ 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.
LA - eng
KW - association schemes; MacWilliams-type identity; weak dual orbit pair; generalized Niederreiter-Rosenbloom-Tsfasman poset
UR - http://eudml.org/doc/285957
ER -