Best Known (89, 89+28, s)-Nets in Base 3
(89, 89+28, 400)-Net over F3 — Constructive and digital
Digital (89, 117, 400)-net over F3, using
- 31 times duplication [i] based on digital (88, 116, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 29, 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, 29, 100)-net over F81, using
(89, 89+28, 687)-Net over F3 — Digital
Digital (89, 117, 687)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3117, 687, F3, 28) (dual of [687, 570, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3117, 757, F3, 28) (dual of [757, 640, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(21) [i] based on
- linear OA(3109, 729, F3, 28) (dual of [729, 620, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(385, 729, F3, 22) (dual of [729, 644, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(27) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3117, 757, F3, 28) (dual of [757, 640, 29]-code), using
(89, 89+28, 29349)-Net in Base 3 — Upper bound on s
There is no (89, 117, 29350)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 66 576434 571097 561461 739680 200355 695856 240614 508328 659621 > 3117 [i]