Best Known (185−35, 185, s)-Nets in Base 3
(185−35, 185, 692)-Net over F3 — Constructive and digital
Digital (150, 185, 692)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 17, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (133, 168, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 42, 172)-net over F81, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 7 and N(F) ≥ 172, using
- net from sequence [i] based on digital (7, 171)-sequence over F81, using
- trace code for nets [i] based on digital (7, 42, 172)-net over F81, using
- digital (0, 17, 4)-net over F3, using
(185−35, 185, 3284)-Net over F3 — Digital
Digital (150, 185, 3284)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3185, 3284, F3, 2, 35) (dual of [(3284, 2), 6383, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3185, 6568, F3, 35) (dual of [6568, 6383, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3185, 6569, F3, 35) (dual of [6569, 6384, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- linear OA(3185, 6561, F3, 35) (dual of [6561, 6376, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3177, 6561, F3, 34) (dual of [6561, 6384, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3185, 6569, F3, 35) (dual of [6569, 6384, 36]-code), using
- OOA 2-folding [i] based on linear OA(3185, 6568, F3, 35) (dual of [6568, 6383, 36]-code), using
(185−35, 185, 523645)-Net in Base 3 — Upper bound on s
There is no (150, 185, 523646)-net in base 3, because
- 1 times m-reduction [i] would yield (150, 184, 523646)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 6170 444903 264727 198924 994465 897624 348746 598087 537947 931102 125208 294772 426069 371012 799709 > 3184 [i]