Best Known (85, 85+20, s)-Nets in Base 3
(85, 85+20, 656)-Net over F3 — Constructive and digital
Digital (85, 105, 656)-net over F3, using
- net defined by OOA [i] based on linear OOA(3105, 656, F3, 20, 20) (dual of [(656, 20), 13015, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(3105, 6560, F3, 20) (dual of [6560, 6455, 21]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(3105, 6561, F3, 20) (dual of [6561, 6456, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(3105, 6560, F3, 20) (dual of [6560, 6455, 21]-code), using
(85, 85+20, 2775)-Net over F3 — Digital
Digital (85, 105, 2775)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3105, 2775, F3, 2, 20) (dual of [(2775, 2), 5445, 21]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3105, 3284, F3, 2, 20) (dual of [(3284, 2), 6463, 21]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3105, 6568, F3, 20) (dual of [6568, 6463, 21]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(3105, 6569, F3, 20) (dual of [6569, 6464, 21]-code), using
- OOA 2-folding [i] based on linear OA(3105, 6568, F3, 20) (dual of [6568, 6463, 21]-code), using
- discarding factors / shortening the dual code based on linear OOA(3105, 3284, F3, 2, 20) (dual of [(3284, 2), 6463, 21]-NRT-code), using
(85, 85+20, 231580)-Net in Base 3 — Upper bound on s
There is no (85, 105, 231581)-net in base 3, because
- the generalized Rao bound for nets shows that 3m ≥ 125 240364 912817 624338 274131 650560 150958 618640 548961 > 3105 [i]