Best Known (84−10, 84, s)-Nets in Base 3
(84−10, 84, 318870)-Net over F3 — Constructive and digital
Digital (74, 84, 318870)-net over F3, using
- 31 times duplication [i] based on digital (73, 83, 318870)-net over F3, using
- net defined by OOA [i] based on linear OOA(383, 318870, F3, 10, 10) (dual of [(318870, 10), 3188617, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(383, 1594350, F3, 10) (dual of [1594350, 1594267, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(383, 1594353, F3, 10) (dual of [1594353, 1594270, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 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(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(383, 1594353, F3, 10) (dual of [1594353, 1594270, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(383, 1594350, F3, 10) (dual of [1594350, 1594267, 11]-code), using
- net defined by OOA [i] based on linear OOA(383, 318870, F3, 10, 10) (dual of [(318870, 10), 3188617, 11]-NRT-code), using
(84−10, 84, 656175)-Net over F3 — Digital
Digital (74, 84, 656175)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(384, 656175, F3, 2, 10) (dual of [(656175, 2), 1312266, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(384, 797177, F3, 2, 10) (dual of [(797177, 2), 1594270, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(384, 1594354, F3, 10) (dual of [1594354, 1594270, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(383, 1594353, F3, 10) (dual of [1594353, 1594270, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- 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(353, 1594323, F3, 7) (dual of [1594323, 1594270, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 1594322 = 313−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [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(9) ⊂ Ce(6) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(383, 1594353, F3, 10) (dual of [1594353, 1594270, 11]-code), using
- OOA 2-folding [i] based on linear OA(384, 1594354, F3, 10) (dual of [1594354, 1594270, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(384, 797177, F3, 2, 10) (dual of [(797177, 2), 1594270, 11]-NRT-code), using
(84−10, 84, large)-Net in Base 3 — Upper bound on s
There is no (74, 84, large)-net in base 3, because
- 8 times m-reduction [i] would yield (74, 76, large)-net in base 3, but