Best Known (106−20, 106, s)-Nets in Base 3
(106−20, 106, 657)-Net over F3 — Constructive and digital
Digital (86, 106, 657)-net over F3, using
- net defined by OOA [i] based on linear OOA(3106, 657, F3, 20, 20) (dual of [(657, 20), 13034, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3106, 6570, F3, 20) (dual of [6570, 6464, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3106, 6570, F3, 20) (dual of [6570, 6464, 21]-code), using
(106−20, 106, 2961)-Net over F3 — Digital
Digital (86, 106, 2961)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3106, 2961, F3, 2, 20) (dual of [(2961, 2), 5816, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3106, 3285, F3, 2, 20) (dual of [(3285, 2), 6464, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3106, 6570, F3, 20) (dual of [6570, 6464, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(397, 6561, F3, 19) (dual of [6561, 6464, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 38−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- OOA 2-folding [i] based on linear OA(3106, 6570, F3, 20) (dual of [6570, 6464, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3106, 3285, F3, 2, 20) (dual of [(3285, 2), 6464, 21]-NRT-code), using
(106−20, 106, 258473)-Net in Base 3 — Upper bound on s
There is no (86, 106, 258474)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 375 720747 336596 403728 734437 134574 119319 200995 726101 > 3106 [i]