Best Known (67, 67+20, s)-Nets in Base 3
(67, 67+20, 400)-Net over F3 — Constructive and digital
Digital (67, 87, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (67, 88, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 22, 100)-net over F81, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 1 and N(F) ≥ 100, using
- net from sequence [i] based on digital (1, 99)-sequence over F81, using
- trace code for nets [i] based on digital (1, 22, 100)-net over F81, using
(67, 67+20, 703)-Net over F3 — Digital
Digital (67, 87, 703)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(387, 703, F3, 20) (dual of [703, 616, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(387, 757, F3, 20) (dual of [757, 670, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- linear OA(379, 729, F3, 20) (dual of [729, 650, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(355, 729, F3, 14) (dual of [729, 674, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- construction X applied to Ce(19) ⊂ Ce(13) [i] based on
- discarding factors / shortening the dual code based on linear OA(387, 757, F3, 20) (dual of [757, 670, 21]-code), using
(67, 67+20, 32045)-Net in Base 3 — Upper bound on s
There is no (67, 87, 32046)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 323267 703099 218689 089605 375997 766751 412781 > 387 [i]