Best Known (59−10, 59, s)-Nets in Base 3
(59−10, 59, 3941)-Net over F3 — Constructive and digital
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
(59−10, 59, 9852)-Net over F3 — Digital
Digital (49, 59, 9852)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(359, 9852, F3, 2, 10) (dual of [(9852, 2), 19645, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(359, 19704, F3, 10) (dual of [19704, 19645, 11]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(359, 19705, F3, 10) (dual of [19705, 19646, 11]-code), using
- OOA 2-folding [i] based on linear OA(359, 19704, F3, 10) (dual of [19704, 19645, 11]-code), using
(59−10, 59, 555691)-Net in Base 3 — Upper bound on s
There is no (49, 59, 555692)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 14130 484056 495551 767272 264249 > 359 [i]