Best Known (210−26, 210, s)-Nets in Base 3
(210−26, 210, 40882)-Net over F3 — Constructive and digital
Digital (184, 210, 40882)-net over F3, using
- 31 times duplication [i] based on digital (183, 209, 40882)-net over F3, using
- net defined by OOA [i] based on linear OOA(3209, 40882, F3, 26, 26) (dual of [(40882, 26), 1062723, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(3209, 531466, F3, 26) (dual of [531466, 531257, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3209, 531469, F3, 26) (dual of [531469, 531260, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3209, 531469, F3, 26) (dual of [531469, 531260, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(3209, 531466, F3, 26) (dual of [531466, 531257, 27]-code), using
- net defined by OOA [i] based on linear OOA(3209, 40882, F3, 26, 26) (dual of [(40882, 26), 1062723, 27]-NRT-code), using
(210−26, 210, 139648)-Net over F3 — Digital
Digital (184, 210, 139648)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3210, 139648, F3, 3, 26) (dual of [(139648, 3), 418734, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3210, 177156, F3, 3, 26) (dual of [(177156, 3), 531258, 27]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3209, 177156, F3, 3, 26) (dual of [(177156, 3), 531259, 27]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3209, 531468, F3, 26) (dual of [531468, 531259, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(3209, 531469, F3, 26) (dual of [531469, 531260, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(22) [i] based on
- linear OA(3205, 531441, F3, 26) (dual of [531441, 531236, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(3181, 531441, F3, 23) (dual of [531441, 531260, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(25) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(3209, 531469, F3, 26) (dual of [531469, 531260, 27]-code), using
- OOA 3-folding [i] based on linear OA(3209, 531468, F3, 26) (dual of [531468, 531259, 27]-code), using
- 31 times duplication [i] based on linear OOA(3209, 177156, F3, 3, 26) (dual of [(177156, 3), 531259, 27]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3210, 177156, F3, 3, 26) (dual of [(177156, 3), 531258, 27]-NRT-code), using
(210−26, 210, large)-Net in Base 3 — Upper bound on s
There is no (184, 210, large)-net in base 3, because
- 24 times m-reduction [i] would yield (184, 186, large)-net in base 3, but