Best Known (146, 146+32, s)-Nets in Base 3
(146, 146+32, 702)-Net over F3 — Constructive and digital
Digital (146, 178, 702)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (6, 22, 14)-net over F3, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 6 and N(F) ≥ 14, using
- net from sequence [i] based on digital (6, 13)-sequence over F3, using
- digital (124, 156, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 39, 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, 39, 172)-net over F81, using
- digital (6, 22, 14)-net over F3, using
(146, 146+32, 3905)-Net over F3 — Digital
Digital (146, 178, 3905)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3178, 3905, F3, 32) (dual of [3905, 3727, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3178, 6596, F3, 32) (dual of [6596, 6418, 33]-code), using
- construction XX applied to Ce(31) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- linear OA(3169, 6561, F3, 32) (dual of [6561, 6392, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(3145, 6561, F3, 28) (dual of [6561, 6416, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3137, 6561, F3, 26) (dual of [6561, 6424, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(36, 32, F3, 3) (dual of [32, 26, 4]-code or 32-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, 3, F3, 1) (dual of [3, 2, 2]-code), using
- Reed–Solomon code RS(2,3) [i]
- construction XX applied to Ce(31) ⊂ Ce(27) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3178, 6596, F3, 32) (dual of [6596, 6418, 33]-code), using
(146, 146+32, 690991)-Net in Base 3 — Upper bound on s
There is no (146, 178, 690992)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 8 464172 644556 541063 088095 234361 619375 075775 634570 675511 836346 824848 866625 757786 655745 > 3178 [i]