Best Known (190−40, 190, s)-Nets in Base 3
(190−40, 190, 688)-Net over F3 — Constructive and digital
Digital (150, 190, 688)-net over F3, using
- 32 times duplication [i] based on digital (148, 188, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 47, 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, 47, 172)-net over F81, using
(190−40, 190, 1738)-Net over F3 — Digital
Digital (150, 190, 1738)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3190, 1738, F3, 40) (dual of [1738, 1548, 41]-code), using
- discarding factors / shortening the dual code based on linear OA(3190, 2214, F3, 40) (dual of [2214, 2024, 41]-code), using
- construction X applied to Ce(39) ⊂ Ce(34) [i] based on
- linear OA(3183, 2187, F3, 40) (dual of [2187, 2004, 41]-code), using an extension Ce(39) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,39], and designed minimum distance d ≥ |I|+1 = 40 [i]
- linear OA(3162, 2187, F3, 35) (dual of [2187, 2025, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(37, 27, F3, 4) (dual of [27, 20, 5]-code), using
- an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- construction X applied to Ce(39) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(3190, 2214, F3, 40) (dual of [2214, 2024, 41]-code), using
(190−40, 190, 141536)-Net in Base 3 — Upper bound on s
There is no (150, 190, 141537)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4 498539 620276 786901 697757 287484 408655 041002 141014 250513 401483 507594 426686 733381 357016 861329 > 3190 [i]