Best Known (73, 73+183, s)-Nets in Base 4
(73, 73+183, 104)-Net over F4 — Constructive and digital
Digital (73, 256, 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
(73, 73+183, 112)-Net over F4 — Digital
Digital (73, 256, 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
(73, 73+183, 448)-Net over F4 — Upper bound on s (digital)
There is no digital (73, 256, 449)-net over F4, because
- 3 times m-reduction [i] would yield digital (73, 253, 449)-net over F4, but
- extracting embedded orthogonal array [i] would yield linear OA(4253, 449, F4, 180) (dual of [449, 196, 181]-code), but
- residual code [i] would yield OA(473, 268, S4, 45), but
- the linear programming bound shows that M ≥ 472 872631 524693 616571 263304 632879 333287 600071 846650 492027 623804 614698 703694 083741 742382 499726 622720 / 5 268581 092830 135391 831262 242618 253418 847780 919324 019929 > 473 [i]
- residual code [i] would yield OA(473, 268, S4, 45), but
- extracting embedded orthogonal array [i] would yield linear OA(4253, 449, F4, 180) (dual of [449, 196, 181]-code), but
(73, 73+183, 490)-Net in Base 4 — Upper bound on s
There is no (73, 256, 491)-net in base 4, because
- 1 times m-reduction [i] would yield (73, 255, 491)-net in base 4, but
- the generalized Rao bound for nets shows that 4m ≥ 3672 242400 578487 759513 078012 925400 531116 460766 559439 139166 443401 101933 932419 399011 536098 143151 422311 477982 273155 808844 169239 004111 004271 925168 347097 468568 > 4255 [i]