Best Known (153−38, 153, s)-Nets in Base 3
(153−38, 153, 328)-Net over F3 — Constructive and digital
Digital (115, 153, 328)-net over F3, using
- 31 times duplication [i] based on digital (114, 152, 328)-net over F3, using
- trace code for nets [i] based on digital (0, 38, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- trace code for nets [i] based on digital (0, 38, 82)-net over F81, using
(153−38, 153, 707)-Net over F3 — Digital
Digital (115, 153, 707)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3153, 707, F3, 38) (dual of [707, 554, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(3153, 749, F3, 38) (dual of [749, 596, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- linear OA(3148, 729, F3, 38) (dual of [729, 581, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(3130, 729, F3, 34) (dual of [729, 599, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(35, 20, F3, 3) (dual of [20, 15, 4]-code or 20-cap in PG(4,3)), using
- construction X applied to Ce(37) ⊂ Ce(33) [i] based on
- discarding factors / shortening the dual code based on linear OA(3153, 749, F3, 38) (dual of [749, 596, 39]-code), using
(153−38, 153, 27540)-Net in Base 3 — Upper bound on s
There is no (115, 153, 27541)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 9 991463 091385 015650 744127 723525 072320 062919 268870 878085 250615 127071 325675 > 3153 [i]