Best Known (53−10, 53, s)-Nets in Base 3
(53−10, 53, 1316)-Net over F3 — Constructive and digital
Digital (43, 53, 1316)-net over F3, using
- net defined by OOA [i] based on linear OOA(353, 1316, F3, 10, 10) (dual of [(1316, 10), 13107, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(353, 6580, F3, 10) (dual of [6580, 6527, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(353, 6581, F3, 10) (dual of [6581, 6528, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 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(353, 6581, F3, 10) (dual of [6581, 6528, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(353, 6580, F3, 10) (dual of [6580, 6527, 11]-code), using
(53−10, 53, 3290)-Net over F3 — Digital
Digital (43, 53, 3290)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(353, 3290, F3, 2, 10) (dual of [(3290, 2), 6527, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(353, 6580, F3, 10) (dual of [6580, 6527, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(353, 6581, F3, 10) (dual of [6581, 6528, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(349, 6561, F3, 10) (dual of [6561, 6512, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(333, 6561, F3, 7) (dual of [6561, 6528, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 20, F3, 2) (dual of [20, 16, 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(353, 6581, F3, 10) (dual of [6581, 6528, 11]-code), using
- OOA 2-folding [i] based on linear OA(353, 6580, F3, 10) (dual of [6580, 6527, 11]-code), using
(53−10, 53, 148688)-Net in Base 3 — Upper bound on s
There is no (43, 53, 148689)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 19 383341 717503 509735 293587 > 353 [i]