Best Known (62, 62+10, s)-Nets in Base 3
(62, 62+10, 35434)-Net over F3 — Constructive and digital
Digital (62, 72, 35434)-net over F3, using
- 31 times duplication [i] based on digital (61, 71, 35434)-net over F3, using
- net defined by OOA [i] based on linear OOA(371, 35434, F3, 10, 10) (dual of [(35434, 10), 354269, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(371, 177170, F3, 10) (dual of [177170, 177099, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(371, 177173, F3, 10) (dual of [177173, 177102, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(371, 177173, F3, 10) (dual of [177173, 177102, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(371, 177170, F3, 10) (dual of [177170, 177099, 11]-code), using
- net defined by OOA [i] based on linear OOA(371, 35434, F3, 10, 10) (dual of [(35434, 10), 354269, 11]-NRT-code), using
(62, 62+10, 88587)-Net over F3 — Digital
Digital (62, 72, 88587)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(372, 88587, F3, 2, 10) (dual of [(88587, 2), 177102, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(372, 177174, F3, 10) (dual of [177174, 177102, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(371, 177173, F3, 10) (dual of [177173, 177102, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(6) [i] based on
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(345, 177147, F3, 7) (dual of [177147, 177102, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(34, 26, F3, 2) (dual of [26, 22, 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(371, 177173, F3, 10) (dual of [177173, 177102, 11]-code), using
- OOA 2-folding [i] based on linear OA(372, 177174, F3, 10) (dual of [177174, 177102, 11]-code), using
(62, 62+10, large)-Net in Base 3 — Upper bound on s
There is no (62, 72, large)-net in base 3, because
- 8 times m-reduction [i] would yield (62, 64, large)-net in base 3, but