Best Known (86, 97, s)-Nets in Base 3
(86, 97, 318871)-Net over F3 — Constructive and digital
Digital (86, 97, 318871)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (0, 5, 4)-net over F3, using
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 0 and N(F) ≥ 4, using
- the rational function field F3(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 3)-sequence over F3, using
- digital (81, 92, 318867)-net over F3, using
- net defined by OOA [i] based on linear OOA(392, 318867, F3, 11, 11) (dual of [(318867, 11), 3507445, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(392, 1594336, F3, 11) (dual of [1594336, 1594244, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(379, 1594323, F3, 10) (dual of [1594323, 1594244, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 13, F3, 0) (dual of [13, 13, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- OOA 5-folding and stacking with additional row [i] based on linear OA(392, 1594336, F3, 11) (dual of [1594336, 1594244, 12]-code), using
- net defined by OOA [i] based on linear OOA(392, 318867, F3, 11, 11) (dual of [(318867, 11), 3507445, 12]-NRT-code), using
- digital (0, 5, 4)-net over F3, using
(86, 97, 797177)-Net over F3 — Digital
Digital (86, 97, 797177)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(397, 797177, F3, 2, 11) (dual of [(797177, 2), 1594257, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(397, 1594354, F3, 11) (dual of [1594354, 1594257, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(396, 1594353, F3, 11) (dual of [1594353, 1594257, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(392, 1594323, F3, 11) (dual of [1594323, 1594231, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(366, 1594323, F3, 8) (dual of [1594323, 1594257, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(34, 30, F3, 2) (dual of [30, 26, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(396, 1594353, F3, 11) (dual of [1594353, 1594257, 12]-code), using
- OOA 2-folding [i] based on linear OA(397, 1594354, F3, 11) (dual of [1594354, 1594257, 12]-code), using
(86, 97, large)-Net in Base 3 — Upper bound on s
There is no (86, 97, large)-net in base 3, because
- 9 times m-reduction [i] would yield (86, 88, large)-net in base 3, but