Best Known (90−21, 90, s)-Nets in Base 3
(90−21, 90, 400)-Net over F3 — Constructive and digital
Digital (69, 90, 400)-net over F3, using
- 32 times duplication [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
(90−21, 90, 664)-Net over F3 — Digital
Digital (69, 90, 664)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(390, 664, F3, 21) (dual of [664, 574, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(390, 749, F3, 21) (dual of [749, 659, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 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(367, 729, F3, 17) (dual of [729, 662, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 728 = 36−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(390, 749, F3, 21) (dual of [749, 659, 22]-code), using
(90−21, 90, 39922)-Net in Base 3 — Upper bound on s
There is no (69, 90, 39923)-net in base 3, because
- 1 times m-reduction [i] would yield (69, 89, 39923)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2 909389 210703 471966 284058 491736 777656 130101 > 389 [i]