Best Known (144, 144+32, s)-Nets in Base 3
(144, 144+32, 700)-Net over F3 — Constructive and digital
Digital (144, 176, 700)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (4, 20, 12)-net over F3, using
- net from sequence [i] based on digital (4, 11)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 4 and N(F) ≥ 12, using
- net from sequence [i] based on digital (4, 11)-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 (4, 20, 12)-net over F3, using
(144, 144+32, 3627)-Net over F3 — Digital
Digital (144, 176, 3627)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3176, 3627, F3, 32) (dual of [3627, 3451, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 6592, F3, 32) (dual of [6592, 6416, 33]-code), using
- 1 times code embedding in larger space [i] 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
- 1 times code embedding in larger space [i] based on linear OA(3175, 6591, F3, 32) (dual of [6591, 6416, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 6592, F3, 32) (dual of [6592, 6416, 33]-code), using
(144, 144+32, 602325)-Net in Base 3 — Upper bound on s
There is no (144, 176, 602326)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 940469 927388 245111 548582 999564 261143 375123 756627 488715 163269 799014 452784 232299 904161 > 3176 [i]