Best Known (84−11, 84, s)-Nets in Base 3
(84−11, 84, 35438)-Net over F3 — Constructive and digital
Digital (73, 84, 35438)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 6, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- digital (67, 78, 35431)-net over F3, using
- net defined by OOA [i] based on linear OOA(378, 35431, F3, 11, 11) (dual of [(35431, 11), 389663, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(378, 177156, F3, 11) (dual of [177156, 177078, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(378, 177158, F3, 11) (dual of [177158, 177080, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(378, 177147, F3, 11) (dual of [177147, 177069, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(30, 11, F3, 0) (dual of [11, 11, 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
- discarding factors / shortening the dual code based on linear OA(378, 177158, F3, 11) (dual of [177158, 177080, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(378, 177156, F3, 11) (dual of [177156, 177078, 12]-code), using
- net defined by OOA [i] based on linear OOA(378, 35431, F3, 11, 11) (dual of [(35431, 11), 389663, 12]-NRT-code), using
- digital (1, 6, 7)-net over F3, using
(84−11, 84, 88593)-Net over F3 — Digital
Digital (73, 84, 88593)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(384, 88593, F3, 2, 11) (dual of [(88593, 2), 177102, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(384, 177186, F3, 11) (dual of [177186, 177102, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(378, 177147, F3, 11) (dual of [177147, 177069, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(36, 39, F3, 3) (dual of [39, 33, 4]-code or 39-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- OOA 2-folding [i] based on linear OA(384, 177186, F3, 11) (dual of [177186, 177102, 12]-code), using
(84−11, 84, large)-Net in Base 3 — Upper bound on s
There is no (73, 84, large)-net in base 3, because
- 9 times m-reduction [i] would yield (73, 75, large)-net in base 3, but