Best Known (175−32, 175, s)-Nets in Base 3
(175−32, 175, 698)-Net over F3 — Constructive and digital
Digital (143, 175, 698)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 19, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-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 (3, 19, 10)-net over F3, using
(175−32, 175, 3496)-Net over F3 — Digital
Digital (143, 175, 3496)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3175, 3496, F3, 32) (dual of [3496, 3321, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3175, 6591, F3, 32) (dual of [6591, 6416, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(27) [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(36, 30, F3, 3) (dual of [30, 24, 4]-code or 30-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(31) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3175, 6591, F3, 32) (dual of [6591, 6416, 33]-code), using
(175−32, 175, 562354)-Net in Base 3 — Upper bound on s
There is no (143, 175, 562355)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 313487 935037 222356 797840 974640 627301 925731 165392 489575 945165 687082 488328 908483 359457 > 3175 [i]