Best Known (144−26, 144, s)-Nets in Base 3
(144−26, 144, 688)-Net over F3 — Constructive and digital
Digital (118, 144, 688)-net over F3, using
- 4 times m-reduction [i] based on digital (118, 148, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 37, 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, 37, 172)-net over F81, using
(144−26, 144, 3391)-Net over F3 — Digital
Digital (118, 144, 3391)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3144, 3391, F3, 26) (dual of [3391, 3247, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 6592, F3, 26) (dual of [6592, 6448, 27]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3143, 6591, F3, 26) (dual of [6591, 6448, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- 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(3113, 6561, F3, 22) (dual of [6561, 6448, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(25) ⊂ Ce(21) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3143, 6591, F3, 26) (dual of [6591, 6448, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 6592, F3, 26) (dual of [6592, 6448, 27]-code), using
(144−26, 144, 546262)-Net in Base 3 — Upper bound on s
There is no (118, 144, 546263)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 507 530361 435359 464647 323658 189631 493548 989107 403981 876632 314224 765527 > 3144 [i]