Best Known (135, 159, s)-Nets in Base 3
(135, 159, 1650)-Net over F3 — Constructive and digital
Digital (135, 159, 1650)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 15, 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 (120, 144, 1640)-net over F3, using
- net defined by OOA [i] based on linear OOA(3144, 1640, F3, 24, 24) (dual of [(1640, 24), 39216, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3144, 19680, F3, 24) (dual of [19680, 19536, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 19683, F3, 24) (dual of [19683, 19539, 25]-code), using
- 1 times truncation [i] based on linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(3145, 19684, F3, 25) (dual of [19684, 19539, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(3144, 19683, F3, 24) (dual of [19683, 19539, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3144, 19680, F3, 24) (dual of [19680, 19536, 25]-code), using
- net defined by OOA [i] based on linear OOA(3144, 1640, F3, 24, 24) (dual of [(1640, 24), 39216, 25]-NRT-code), using
- digital (3, 15, 10)-net over F3, using
(135, 159, 12069)-Net over F3 — Digital
Digital (135, 159, 12069)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3159, 12069, F3, 24) (dual of [12069, 11910, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3159, 19736, F3, 24) (dual of [19736, 19577, 25]-code), using
- construction X applied to Ce(24) ⊂ Ce(16) [i] based on
- linear OA(3145, 19683, F3, 25) (dual of [19683, 19538, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(314, 53, F3, 6) (dual of [53, 39, 7]-code), using
- construction X applied to Ce(24) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3159, 19736, F3, 24) (dual of [19736, 19577, 25]-code), using
(135, 159, 5548647)-Net in Base 3 — Upper bound on s
There is no (135, 159, 5548648)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 7282 489708 689492 892181 364088 314110 819355 396958 016427 916970 134729 172922 701377 > 3159 [i]