Best Known (47, 58, s)-Nets in Base 3
(47, 58, 1313)-Net over F3 — Constructive and digital
Digital (47, 58, 1313)-net over F3, using
- 31 times duplication [i] based on digital (46, 57, 1313)-net over F3, using
- net defined by OOA [i] based on linear OOA(357, 1313, F3, 11, 11) (dual of [(1313, 11), 14386, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(357, 6566, F3, 11) (dual of [6566, 6509, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(357, 6569, F3, 11) (dual of [6569, 6512, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(357, 6569, F3, 11) (dual of [6569, 6512, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(357, 6566, F3, 11) (dual of [6566, 6509, 12]-code), using
- net defined by OOA [i] based on linear OOA(357, 1313, F3, 11, 11) (dual of [(1313, 11), 14386, 12]-NRT-code), using
(47, 58, 3285)-Net over F3 — Digital
Digital (47, 58, 3285)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(358, 3285, F3, 2, 11) (dual of [(3285, 2), 6512, 12]-NRT-code), using
- OOA 2-folding [i] based on linear OA(358, 6570, F3, 11) (dual of [6570, 6512, 12]-code), using
- 1 times code embedding in larger space [i] based on linear OA(357, 6569, F3, 11) (dual of [6569, 6512, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(357, 6561, F3, 11) (dual of [6561, 6504, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(30, 8, F3, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(357, 6569, F3, 11) (dual of [6569, 6512, 12]-code), using
- OOA 2-folding [i] based on linear OA(358, 6570, F3, 11) (dual of [6570, 6512, 12]-code), using
(47, 58, 358082)-Net in Base 3 — Upper bound on s
There is no (47, 58, 358083)-net in base 3, because
- 1 times m-reduction [i] would yield (47, 57, 358083)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 1570 053811 289974 741721 136895 > 357 [i]