Best Known (33, 33+34, s)-Nets in Base 81
(33, 33+34, 452)-Net over F81 — Constructive and digital
Digital (33, 67, 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, 50, 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
(33, 33+34, 1760)-Net over F81 — Digital
Digital (33, 67, 1760)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8167, 1760, F81, 3, 34) (dual of [(1760, 3), 5213, 35]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8167, 2187, F81, 3, 34) (dual of [(2187, 3), 6494, 35]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using
- an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- OOA 3-folding [i] based on linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using
- discarding factors / shortening the dual code based on linear OOA(8167, 2187, F81, 3, 34) (dual of [(2187, 3), 6494, 35]-NRT-code), using
(33, 33+34, 2982155)-Net in Base 81 — Upper bound on s
There is no (33, 67, 2982156)-net in base 81, because
- 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]