Best Known (136, 136+34, s)-Nets in Base 3
(136, 136+34, 688)-Net over F3 — Constructive and digital
Digital (136, 170, 688)-net over F3, using
- 2 times m-reduction [i] based on digital (136, 172, 688)-net over F3, using
- trace code for nets [i] based on digital (7, 43, 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, 43, 172)-net over F81, using
(136, 136+34, 2087)-Net over F3 — Digital
Digital (136, 170, 2087)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3170, 2087, F3, 34) (dual of [2087, 1917, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(3170, 2231, F3, 34) (dual of [2231, 2061, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- linear OA(3155, 2187, F3, 34) (dual of [2187, 2032, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(3120, 2187, F3, 26) (dual of [2187, 2067, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 2186 = 37−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(315, 44, F3, 7) (dual of [44, 29, 8]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
- discarding factors / shortening the dual code based on linear OA(3170, 2231, F3, 34) (dual of [2231, 2061, 35]-code), using
(136, 136+34, 211881)-Net in Base 3 — Upper bound on s
There is no (136, 170, 211882)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 1290 076289 606078 822840 981304 540572 658308 932597 206787 658581 501113 442618 655847 860661 > 3170 [i]