Best Known (189, 224, s)-Nets in Base 3
(189, 224, 1490)-Net over F3 — Constructive and digital
Digital (189, 224, 1490)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 20, 10)-net over F3, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 3 and N(F) ≥ 10, using
- net from sequence [i] based on digital (3, 9)-sequence over F3, using
- digital (169, 204, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- trace code for nets [i] based on digital (16, 51, 370)-net over F81, using
- digital (3, 20, 10)-net over F3, using
(189, 224, 10997)-Net over F3 — Digital
Digital (189, 224, 10997)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3224, 10997, F3, 35) (dual of [10997, 10773, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 19744, F3, 35) (dual of [19744, 19520, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- linear OA(3208, 19683, F3, 35) (dual of [19683, 19475, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(316, 61, F3, 6) (dual of [61, 45, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- the primitive narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 7 [i]
- discarding factors / shortening the dual code based on linear OA(316, 80, F3, 6) (dual of [80, 64, 7]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(3224, 19744, F3, 35) (dual of [19744, 19520, 36]-code), using
(189, 224, 6510619)-Net in Base 3 — Upper bound on s
There is no (189, 224, 6510620)-net in base 3, because
- 1 times m-reduction [i] would yield (189, 223, 6510620)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 25005 758792 184611 351286 133697 209229 201443 776892 666646 758200 541090 850003 152351 084635 686882 991472 594760 552505 > 3223 [i]