Best Known (70−41, 70, s)-Nets in Base 81
(70−41, 70, 370)-Net over F81 — Constructive and digital
Digital (29, 70, 370)-net over F81, using
- t-expansion [i] based on digital (16, 70, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
(70−41, 70, 501)-Net over F81 — Digital
Digital (29, 70, 501)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8170, 501, F81, 3, 41) (dual of [(501, 3), 1433, 42]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8167, 500, F81, 3, 41) (dual of [(500, 3), 1433, 42]-NRT-code), using
- extended algebraic-geometric NRT-code AGe(3;F,1458P) [i] based on function field F/F81 with g(F) = 26 and N(F) ≥ 500, using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(8167, 500, F81, 3, 41) (dual of [(500, 3), 1433, 42]-NRT-code), using
(70−41, 70, 398546)-Net in Base 81 — Upper bound on s
There is no (29, 70, 398547)-net in base 81, because
- 1 times m-reduction [i] would yield (29, 69, 398547)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 484706 113747 021993 934441 285354 313960 055259 706357 607572 369078 827338 281473 970700 050857 012410 651258 720418 234764 153995 752608 709601 643201 > 8169 [i]