Best Known (68−19, 68, s)-Nets in Base 3
(68−19, 68, 156)-Net over F3 — Constructive and digital
Digital (49, 68, 156)-net over F3, using
- 1 times m-reduction [i] based on digital (49, 69, 156)-net over F3, using
- trace code for nets [i] based on digital (3, 23, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- trace code for nets [i] based on digital (3, 23, 52)-net over F27, using
(68−19, 68, 259)-Net over F3 — Digital
Digital (49, 68, 259)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(368, 259, F3, 19) (dual of [259, 191, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(368, 265, F3, 19) (dual of [265, 197, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(361, 243, F3, 19) (dual of [243, 182, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(346, 243, F3, 14) (dual of [243, 197, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 242 = 35−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(37, 22, F3, 4) (dual of [22, 15, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 26 = 33−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(37, 26, F3, 4) (dual of [26, 19, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(368, 265, F3, 19) (dual of [265, 197, 20]-code), using
(68−19, 68, 7381)-Net in Base 3 — Upper bound on s
There is no (49, 68, 7382)-net in base 3, because
- 1 times m-reduction [i] would yield (49, 67, 7382)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 92 804524 648572 284169 368929 380125 > 367 [i]