Best Known (84, 84+149, s)-Nets in Base 3
(84, 84+149, 59)-Net over F3 — Constructive and digital
Digital (84, 233, 59)-net over F3, using
- net from sequence [i] based on digital (84, 58)-sequence over F3, using
- base reduction for sequences [i] based on digital (13, 58)-sequence over F9, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F9 with g(F) = 13 and N(F) ≥ 64, using
- s-reduction based on digital (13, 63)-sequence over F9, using
- base reduction for sequences [i] based on digital (13, 58)-sequence over F9, using
(84, 84+149, 84)-Net over F3 — Digital
Digital (84, 233, 84)-net over F3, using
- t-expansion [i] based on digital (71, 233, 84)-net over F3, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 71 and N(F) ≥ 84, using
- net from sequence [i] based on digital (71, 83)-sequence over F3, using
(84, 84+149, 372)-Net over F3 — Upper bound on s (digital)
There is no digital (84, 233, 373)-net over F3, because
- 2 times m-reduction [i] would yield digital (84, 231, 373)-net over F3, but
- extracting embedded orthogonal array [i] would yield linear OA(3231, 373, F3, 147) (dual of [373, 142, 148]-code), but
- residual code [i] would yield linear OA(384, 225, F3, 49) (dual of [225, 141, 50]-code), but
- 1 times truncation [i] would yield linear OA(383, 224, F3, 48) (dual of [224, 141, 49]-code), but
- the Johnson bound shows that N ≤ 18 650275 231410 181575 562142 123676 906294 083572 717553 183697 141026 684466 < 3141 [i]
- 1 times truncation [i] would yield linear OA(383, 224, F3, 48) (dual of [224, 141, 49]-code), but
- residual code [i] would yield linear OA(384, 225, F3, 49) (dual of [225, 141, 50]-code), but
- extracting embedded orthogonal array [i] would yield linear OA(3231, 373, F3, 147) (dual of [373, 142, 148]-code), but
(84, 84+149, 375)-Net in Base 3 — Upper bound on s
There is no (84, 233, 376)-net in base 3, because
- 1 times m-reduction [i] would yield (84, 232, 376)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 530 832722 876684 135527 794554 380622 940706 594698 665075 369067 506417 698033 733257 019724 990478 809762 988142 076597 410769 > 3232 [i]