Best Known (85, 85+26, s)-Nets in Base 3
(85, 85+26, 400)-Net over F3 — Constructive and digital
Digital (85, 111, 400)-net over F3, using
- 1 times m-reduction [i] based on digital (85, 112, 400)-net over F3, using
- trace code for nets [i] based on digital (1, 28, 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, 28, 100)-net over F81, using
(85, 85+26, 732)-Net over F3 — Digital
Digital (85, 111, 732)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3111, 732, F3, 26) (dual of [732, 621, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3111, 757, F3, 26) (dual of [757, 646, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(3103, 729, F3, 26) (dual of [729, 626, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- 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(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(25) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(3111, 757, F3, 26) (dual of [757, 646, 27]-code), using
(85, 85+26, 33581)-Net in Base 3 — Upper bound on s
There is no (85, 111, 33582)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 91326 257685 129737 378121 589370 515246 436564 033382 060261 > 3111 [i]