Best Known (67, 78, s)-Nets in Base 3
(67, 78, 35431)-Net over F3 — Constructive and digital
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
(67, 78, 64160)-Net over F3 — Digital
Digital (67, 78, 64160)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(378, 64160, F3, 2, 11) (dual of [(64160, 2), 128242, 12]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(378, 88579, F3, 2, 11) (dual of [(88579, 2), 177080, 12]-NRT-code), using
- OOA 2-folding [i] 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
- OOA 2-folding [i] based on linear OA(378, 177158, F3, 11) (dual of [177158, 177080, 12]-code), using
- discarding factors / shortening the dual code based on linear OOA(378, 88579, F3, 2, 11) (dual of [(88579, 2), 177080, 12]-NRT-code), using
(67, 78, large)-Net in Base 3 — Upper bound on s
There is no (67, 78, large)-net in base 3, because
- 9 times m-reduction [i] would yield (67, 69, large)-net in base 3, but