Best Known (60, 60+10, s)-Nets in Base 3
(60, 60+10, 35432)-Net over F3 — Constructive and digital
Digital (60, 70, 35432)-net over F3, using
- 32 times duplication [i] based on digital (58, 68, 35432)-net over F3, using
- net defined by OOA [i] based on linear OOA(368, 35432, F3, 10, 10) (dual of [(35432, 10), 354252, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(368, 177160, F3, 10) (dual of [177160, 177092, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [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(356, 177147, F3, 8) (dual of [177147, 177091, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- OA 5-folding and stacking [i] based on linear OA(368, 177160, F3, 10) (dual of [177160, 177092, 11]-code), using
- net defined by OOA [i] based on linear OOA(368, 35432, F3, 10, 10) (dual of [(35432, 10), 354252, 11]-NRT-code), using
(60, 60+10, 72903)-Net over F3 — Digital
Digital (60, 70, 72903)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(370, 72903, F3, 2, 10) (dual of [(72903, 2), 145736, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(370, 88581, F3, 2, 10) (dual of [(88581, 2), 177092, 11]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(368, 88580, F3, 2, 10) (dual of [(88580, 2), 177092, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(368, 177160, F3, 10) (dual of [177160, 177092, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [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(356, 177147, F3, 8) (dual of [177147, 177091, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(312, 13, F3, 12) (dual of [13, 1, 13]-code or 13-arc in PG(11,3)), using
- dual of repetition code with length 13 [i]
- linear OA(31, 13, F3, 1) (dual of [13, 12, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X4 applied to Ce(9) ⊂ Ce(7) [i] based on
- OOA 2-folding [i] based on linear OA(368, 177160, F3, 10) (dual of [177160, 177092, 11]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(368, 88580, F3, 2, 10) (dual of [(88580, 2), 177092, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(370, 88581, F3, 2, 10) (dual of [(88581, 2), 177092, 11]-NRT-code), using
(60, 60+10, 6230221)-Net in Base 3 — Upper bound on s
There is no (60, 70, 6230222)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 2503 155969 238899 337588 430820 494901 > 370 [i]