Best Known (58−10, 58, s)-Nets in Base 3
(58−10, 58, 3939)-Net over F3 — Constructive and digital
Digital (48, 58, 3939)-net over F3, using
- 31 times duplication [i] based on digital (47, 57, 3939)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 3939, F3, 10, 10) (dual of [(3939, 10), 39333, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(357, 19695, F3, 10) (dual of [19695, 19638, 11]-code), using
- 1 times code embedding in larger space [i] based on linear OA(356, 19694, F3, 10) (dual of [19694, 19638, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [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(346, 19683, F3, 8) (dual of [19683, 19637, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(310, 11, F3, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,3)), using
- dual of repetition code with length 11 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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
- 1 times code embedding in larger space [i] based on linear OA(356, 19694, F3, 10) (dual of [19694, 19638, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(357, 19695, F3, 10) (dual of [19695, 19638, 11]-code), using
- net defined by OOA [i] based on linear OOA(357, 3939, F3, 10, 10) (dual of [(3939, 10), 39333, 11]-NRT-code), using
(58−10, 58, 9848)-Net over F3 — Digital
Digital (48, 58, 9848)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(358, 9848, F3, 2, 10) (dual of [(9848, 2), 19638, 11]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(356, 9847, F3, 2, 10) (dual of [(9847, 2), 19638, 11]-NRT-code), using
- OOA 2-folding [i] based on linear OA(356, 19694, F3, 10) (dual of [19694, 19638, 11]-code), using
- construction X4 applied to Ce(9) ⊂ Ce(7) [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(346, 19683, F3, 8) (dual of [19683, 19637, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(310, 11, F3, 10) (dual of [11, 1, 11]-code or 11-arc in PG(9,3)), using
- dual of repetition code with length 11 [i]
- linear OA(31, 11, F3, 1) (dual of [11, 10, 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(356, 19694, F3, 10) (dual of [19694, 19638, 11]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(356, 9847, F3, 2, 10) (dual of [(9847, 2), 19638, 11]-NRT-code), using
(58−10, 58, 446075)-Net in Base 3 — Upper bound on s
There is no (48, 58, 446076)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 4710 152045 401238 238663 238873 > 358 [i]