Best Known (66−15, 66, s)-Nets in Base 3
(66−15, 66, 400)-Net over F3 — Constructive and digital
Digital (51, 66, 400)-net over F3, using
- 32 times duplication [i] based on digital (49, 64, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 16, 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, 16, 100)-net over F81, using
(66−15, 66, 677)-Net over F3 — Digital
Digital (51, 66, 677)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(366, 677, F3, 15) (dual of [677, 611, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(366, 749, F3, 15) (dual of [749, 683, 16]-code), using
- construction X applied to Ce(15) ⊂ Ce(10) [i] based on
- linear OA(361, 729, F3, 16) (dual of [729, 668, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(343, 729, F3, 11) (dual of [729, 686, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(15) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(366, 749, F3, 15) (dual of [749, 683, 16]-code), using
(66−15, 66, 45523)-Net in Base 3 — Upper bound on s
There is no (51, 66, 45524)-net in base 3, because
- 1 times m-reduction [i] would yield (51, 65, 45524)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 10 301402 215240 544614 387451 969457 > 365 [i]