Best Known (258−185, 258, s)-Nets in Base 4
(258−185, 258, 104)-Net over F4 — Constructive and digital
Digital (73, 258, 104)-net over F4, using
- net from sequence [i] based on digital (73, 103)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 104, using
- F6 from the tower of function fields by GarcÃa and Stichtenoth over F4 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 104, using
(258−185, 258, 112)-Net over F4 — Digital
Digital (73, 258, 112)-net over F4, using
- net from sequence [i] based on digital (73, 111)-sequence over F4, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F4 with g(F) = 73 and N(F) ≥ 112, using
(258−185, 258, 424)-Net over F4 — Upper bound on s (digital)
There is no digital (73, 258, 425)-net over F4, because
- 1 times m-reduction [i] would yield digital (73, 257, 425)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4257, 425, F4, 184) (dual of [425, 168, 185]-code), but
- residual code [i] would yield OA(473, 240, S4, 46), but
- the linear programming bound shows that M ≥ 3081 664986 883081 627986 607924 585409 348828 403557 841996 916067 930922 037066 224815 129884 411985 660277 287813 120000 / 34 388507 451119 783895 235565 346914 908818 017904 036178 247038 198799 > 473 [i]
- residual code [i] would yield OA(473, 240, S4, 46), but
- extracting embedded orthogonal array [i] would yield linear OA(4257, 425, F4, 184) (dual of [425, 168, 185]-code), but
(258−185, 258, 488)-Net in Base 4 — Upper bound on s
There is no (73, 258, 489)-net in base 4, because
- 1 times m-reduction [i] would yield (73, 257, 489)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 54904 500512 186253 385738 848773 880191 537109 898203 002688 021087 462629 558701 640714 860187 160289 369131 309415 346734 996071 401914 174362 703622 740850 551206 750701 981320 > 4257 [i]