Best Known (188−29, 188, s)-Nets in Base 3
(188−29, 188, 1480)-Net over F3 — Constructive and digital
Digital (159, 188, 1480)-net over F3, using
- t-expansion [i] based on digital (157, 188, 1480)-net over F3, using
- trace code for nets [i] based on digital (16, 47, 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, 47, 370)-net over F81, using
(188−29, 188, 10987)-Net over F3 — Digital
Digital (159, 188, 10987)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3188, 10987, F3, 29) (dual of [10987, 10799, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(3188, 19744, F3, 29) (dual of [19744, 19556, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(21) [i] based on
- linear OA(3172, 19683, F3, 29) (dual of [19683, 19511, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(28) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3188, 19744, F3, 29) (dual of [19744, 19556, 30]-code), using
(188−29, 188, 7135159)-Net in Base 3 — Upper bound on s
There is no (159, 188, 7135160)-net in base 3, because
- 1 times m-reduction [i] would yield (159, 187, 7135160)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 166600 046599 147804 415697 324648 486536 464789 395089 710364 314786 815561 534991 571573 510533 564881 > 3187 [i]