Best Known (190, 240, s)-Nets in Base 3
(190, 240, 688)-Net over F3 — Constructive and digital
Digital (190, 240, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (190, 244, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 61, 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, 61, 172)-net over F81, using
(190, 240, 2181)-Net over F3 — Digital
Digital (190, 240, 2181)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3240, 2181, F3, 50) (dual of [2181, 1941, 51]-code), using
- discarding factors / shortening the dual code based on linear OA(3240, 2217, F3, 50) (dual of [2217, 1977, 51]-code), using
- construction XX applied to Ce(49) ⊂ Ce(45) ⊂ Ce(43) [i] based on
- linear OA(3232, 2187, F3, 50) (dual of [2187, 1955, 51]-code), using an extension Ce(49) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,49], and designed minimum distance d ≥ |I|+1 = 50 [i]
- linear OA(3211, 2187, F3, 46) (dual of [2187, 1976, 47]-code), using an extension Ce(45) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,45], and designed minimum distance d ≥ |I|+1 = 46 [i]
- linear OA(3204, 2187, F3, 44) (dual of [2187, 1983, 45]-code), using an extension Ce(43) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,43], and designed minimum distance d ≥ |I|+1 = 44 [i]
- linear OA(36, 28, F3, 3) (dual of [28, 22, 4]-code or 28-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- linear OA(31, 2, F3, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(49) ⊂ Ce(45) ⊂ Ce(43) [i] based on
- discarding factors / shortening the dual code based on linear OA(3240, 2217, F3, 50) (dual of [2217, 1977, 51]-code), using
(190, 240, 193599)-Net in Base 3 — Upper bound on s
There is no (190, 240, 193600)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 3 229325 873151 826858 605414 517432 286397 146210 754989 663929 335695 029603 944421 001032 558491 885352 552474 119704 682150 036609 > 3240 [i]