Best Known (65−33, 65, s)-Nets in Base 81
(65−33, 65, 452)-Net over F81 — Constructive and digital
Digital (32, 65, 452)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (0, 16, 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, 49, 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, 16, 82)-net over F81, using
(65−33, 65, 1741)-Net over F81 — Digital
Digital (32, 65, 1741)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8165, 1741, F81, 3, 33) (dual of [(1741, 3), 5158, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8165, 2187, F81, 3, 33) (dual of [(2187, 3), 6496, 34]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8165, 6561, F81, 33) (dual of [6561, 6496, 34]-code), using
- an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- OOA 3-folding [i] based on linear OA(8165, 6561, F81, 33) (dual of [6561, 6496, 34]-code), using
- discarding factors / shortening the dual code based on linear OOA(8165, 2187, F81, 3, 33) (dual of [(2187, 3), 6496, 34]-NRT-code), using
(65−33, 65, 3659214)-Net in Base 81 — Upper bound on s
There is no (32, 65, 3659215)-net in base 81, because
- 1 times m-reduction [i] would yield (32, 64, 3659215)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 139 008597 227023 906161 048557 981766 458611 197789 288373 488154 565008 348165 209739 390535 797107 401873 786636 053333 572953 712705 555201 > 8164 [i]