Best Known (146, 146+38, s)-Nets in Base 3
(146, 146+38, 688)-Net over F3 — Constructive and digital
Digital (146, 184, 688)-net over F3, using
- t-expansion [i] based on digital (145, 184, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 46, 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, 46, 172)-net over F81, using
(146, 146+38, 1868)-Net over F3 — Digital
Digital (146, 184, 1868)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3184, 1868, F3, 38) (dual of [1868, 1684, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3184, 2217, F3, 38) (dual of [2217, 2033, 39]-code), using
- construction XX applied to Ce(37) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- linear OA(3176, 2187, F3, 38) (dual of [2187, 2011, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3148, 2187, F3, 32) (dual of [2187, 2039, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [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(37) ⊂ Ce(33) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(3184, 2217, F3, 38) (dual of [2217, 2033, 39]-code), using
(146, 146+38, 165454)-Net in Base 3 — Upper bound on s
There is no (146, 184, 165455)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 6170 436099 689081 437158 802220 719059 057222 777956 845731 406996 643269 692085 844033 881169 492507 > 3184 [i]