Best Known (69−35, 69, s)-Nets in Base 81
(69−35, 69, 470)-Net over F81 — Constructive and digital
Digital (34, 69, 470)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (1, 18, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-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 (1, 18, 100)-net over F81, using
(69−35, 69, 1781)-Net over F81 — Digital
Digital (34, 69, 1781)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8169, 1781, F81, 3, 35) (dual of [(1781, 3), 5274, 36]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8169, 2187, F81, 3, 35) (dual of [(2187, 3), 6492, 36]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8169, 6561, F81, 35) (dual of [6561, 6492, 36]-code), using
- an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- OOA 3-folding [i] based on linear OA(8169, 6561, F81, 35) (dual of [6561, 6492, 36]-code), using
- discarding factors / shortening the dual code based on linear OOA(8169, 2187, F81, 3, 35) (dual of [(2187, 3), 6492, 36]-NRT-code), using
(69−35, 69, 3861840)-Net in Base 81 — Upper bound on s
There is no (34, 69, 3861841)-net in base 81, because
- 1 times m-reduction [i] would yield (34, 68, 3861841)-net in base 81, but
- the generalized Rao bound for nets shows that 81m ≥ 5983 862111 232974 397531 166581 232677 658397 698634 819449 865602 319999 983599 009623 634224 730557 720000 814205 421983 563361 715413 355925 358161 > 8168 [i]