Best Known (78, 78+11, s)-Nets in Base 3
(78, 78+11, 106293)-Net over F3 — Constructive and digital
Digital (78, 89, 106293)-net over F3, using
- net defined by OOA [i] based on linear OOA(389, 106293, F3, 11, 11) (dual of [(106293, 11), 1169134, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(389, 531466, F3, 11) (dual of [531466, 531377, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 531469, F3, 11) (dual of [531469, 531380, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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
- discarding factors / shortening the dual code based on linear OA(389, 531469, F3, 11) (dual of [531469, 531380, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(389, 531466, F3, 11) (dual of [531466, 531377, 12]-code), using
(78, 78+11, 265734)-Net over F3 — Digital
Digital (78, 89, 265734)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(389, 265734, F3, 2, 11) (dual of [(265734, 2), 531379, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(389, 531468, F3, 11) (dual of [531468, 531379, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(389, 531469, F3, 11) (dual of [531469, 531380, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(7) [i] based on
- linear OA(385, 531441, F3, 11) (dual of [531441, 531356, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(361, 531441, F3, 8) (dual of [531441, 531380, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(34, 28, F3, 2) (dual of [28, 24, 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
- discarding factors / shortening the dual code based on linear OA(389, 531469, F3, 11) (dual of [531469, 531380, 12]-code), using
- OOA 2-folding [i] based on linear OA(389, 531468, F3, 11) (dual of [531468, 531379, 12]-code), using
(78, 78+11, large)-Net in Base 3 — Upper bound on s
There is no (78, 89, large)-net in base 3, because
- 9 times m-reduction [i] would yield (78, 80, large)-net in base 3, but