Best Known (50, 50+10, s)-Nets in Base 3
(50, 50+10, 3941)-Net over F3 — Constructive and digital
Digital (50, 60, 3941)-net over F3, using
- 31 times duplication [i] based on digital (49, 59, 3941)-net over F3, using
- net defined by OOA [i] based on linear OOA(359, 3941, F3, 10, 10) (dual of [(3941, 10), 39351, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(359, 19705, F3, 10) (dual of [19705, 19646, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(337, 19683, F3, 7) (dual of [19683, 19646, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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
- OA 5-folding and stacking [i] based on linear OA(359, 19705, F3, 10) (dual of [19705, 19646, 11]-code), using
- net defined by OOA [i] based on linear OOA(359, 3941, F3, 10, 10) (dual of [(3941, 10), 39351, 11]-NRT-code), using
(50, 50+10, 9853)-Net over F3 — Digital
Digital (50, 60, 9853)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(360, 9853, F3, 2, 10) (dual of [(9853, 2), 19646, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(360, 19706, F3, 10) (dual of [19706, 19646, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(359, 19705, F3, 10) (dual of [19705, 19646, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(355, 19683, F3, 10) (dual of [19683, 19628, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(337, 19683, F3, 7) (dual of [19683, 19646, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 22, F3, 2) (dual of [22, 18, 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(359, 19705, F3, 10) (dual of [19705, 19646, 11]-code), using
- OOA 2-folding [i] based on linear OA(360, 19706, F3, 10) (dual of [19706, 19646, 11]-code), using
(50, 50+10, 692242)-Net in Base 3 — Upper bound on s
There is no (50, 60, 692243)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 42391 200154 120740 814633 875999 > 360 [i]