Best Known (68−35, 68, s)-Nets in Base 81
(68−35, 68, 452)-Net over F81 — Constructive and digital
Digital (33, 68, 452)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- digital (16, 51, 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
- digital (0, 17, 82)-net over F81, using
(68−35, 68, 1218)-Net over F81 — Digital
Digital (33, 68, 1218)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8168, 1218, F81, 35) (dual of [1218, 1150, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(8168, 1313, F81, 35) (dual of [1313, 1245, 36]-code), using
(68−35, 68, 2982155)-Net in Base 81 — Upper bound on s
There is no (33, 68, 2982156)-net in base 81, because
- 1 times m-reduction [i] would yield (33, 67, 2982156)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 73 875032 370870 585493 305764 167764 483812 640979 313804 527044 176764 461900 384991 782574 634700 150782 376021 654214 894982 474926 057840 410561 > 8167 [i]