Best Known (188−30, 188, s)-Nets in Base 3
(188−30, 188, 1480)-Net over F3 — Constructive and digital
Digital (158, 188, 1480)-net over F3, using
- t-expansion [i] based on digital (157, 188, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 47, 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
- trace code for nets [i] based on digital (16, 47, 370)-net over F81, using
(188−30, 188, 9858)-Net over F3 — Digital
Digital (158, 188, 9858)-net over F3, using
- 31 times duplication [i] based on digital (157, 187, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 9858, F3, 2, 30) (dual of [(9858, 2), 19529, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3187, 19716, F3, 30) (dual of [19716, 19529, 31]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-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
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3187, 19716, F3, 30) (dual of [19716, 19529, 31]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3187, 9858, F3, 2, 30) (dual of [(9858, 2), 19529, 31]-NRT-code), using
(188−30, 188, 3066580)-Net in Base 3 — Upper bound on s
There is no (158, 188, 3066581)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 499801 706288 689930 105578 125489 571844 679891 516712 800961 263975 683968 188954 198571 323694 550971 > 3188 [i]