Best Known (190−31, 190, s)-Nets in Base 3
(190−31, 190, 1480)-Net over F3 — Constructive and digital
Digital (159, 190, 1480)-net over F3, using
- 32 times duplication [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
(190−31, 190, 8999)-Net over F3 — Digital
Digital (159, 190, 8999)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3190, 8999, F3, 2, 31) (dual of [(8999, 2), 17808, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3190, 9859, F3, 2, 31) (dual of [(9859, 2), 19528, 32]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3189, 9859, F3, 2, 31) (dual of [(9859, 2), 19529, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3189, 19718, F3, 31) (dual of [19718, 19529, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- linear OA(3181, 19683, F3, 31) (dual of [19683, 19502, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3154, 19683, F3, 26) (dual of [19683, 19529, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(38, 35, F3, 4) (dual of [35, 27, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- the narrow-sense BCH-code C(I) with length 41 | 38−1, defining interval I = [1,1], and minimum distance d ≥ |{−3,−1,1,3}|+1 = 5 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(38, 41, F3, 4) (dual of [41, 33, 5]-code), using
- construction X applied to Ce(30) ⊂ Ce(25) [i] based on
- OOA 2-folding [i] based on linear OA(3189, 19718, F3, 31) (dual of [19718, 19529, 32]-code), using
- 31 times duplication [i] based on linear OOA(3189, 9859, F3, 2, 31) (dual of [(9859, 2), 19529, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3190, 9859, F3, 2, 31) (dual of [(9859, 2), 19528, 32]-NRT-code), using
(190−31, 190, 3299609)-Net in Base 3 — Upper bound on s
There is no (159, 190, 3299610)-net in base 3, because
- 1 times m-reduction [i] would yield (159, 189, 3299610)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1 499402 111366 350671 838755 124210 874171 076164 918136 992046 193986 720475 724037 205440 201367 211961 > 3189 [i]