Best Known (214−27, 214, s)-Nets in Base 3
(214−27, 214, 13636)-Net over F3 — Constructive and digital
Digital (187, 214, 13636)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 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 (171, 198, 13626)-net over F3, using
- net defined by OOA [i] based on linear OOA(3198, 13626, F3, 27, 27) (dual of [(13626, 27), 367704, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3198, 177139, F3, 27) (dual of [177139, 176941, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 177146, F3, 27) (dual of [177146, 176948, 28]-code), using
- 1 times truncation [i] based on linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3198, 177146, F3, 27) (dual of [177146, 176948, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3198, 177139, F3, 27) (dual of [177139, 176941, 28]-code), using
- net defined by OOA [i] based on linear OOA(3198, 13626, F3, 27, 27) (dual of [(13626, 27), 367704, 28]-NRT-code), using
- digital (3, 16, 10)-net over F3, using
(214−27, 214, 80312)-Net over F3 — Digital
Digital (187, 214, 80312)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3214, 80312, F3, 2, 27) (dual of [(80312, 2), 160410, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3214, 88608, F3, 2, 27) (dual of [(88608, 2), 177002, 28]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3214, 177216, F3, 27) (dual of [177216, 177002, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 177217, F3, 27) (dual of [177217, 177003, 28]-code), using
- 1 times truncation [i] based on linear OA(3215, 177218, F3, 28) (dual of [177218, 177003, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- linear OA(3199, 177147, F3, 28) (dual of [177147, 176948, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- linear OA(3144, 177147, F3, 20) (dual of [177147, 177003, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(316, 71, F3, 7) (dual of [71, 55, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(316, 82, F3, 7) (dual of [82, 66, 8]-code), using
- construction X applied to Ce(27) ⊂ Ce(19) [i] based on
- 1 times truncation [i] based on linear OA(3215, 177218, F3, 28) (dual of [177218, 177003, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3214, 177217, F3, 27) (dual of [177217, 177003, 28]-code), using
- OOA 2-folding [i] based on linear OA(3214, 177216, F3, 27) (dual of [177216, 177002, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(3214, 88608, F3, 2, 27) (dual of [(88608, 2), 177002, 28]-NRT-code), using
(214−27, 214, large)-Net in Base 3 — Upper bound on s
There is no (187, 214, large)-net in base 3, because
- 25 times m-reduction [i] would yield (187, 189, large)-net in base 3, but