Best Known (70−20, 70, s)-Nets in Base 9
(70−20, 70, 656)-Net over F9 — Constructive and digital
Digital (50, 70, 656)-net over F9, using
- 91 times duplication [i] based on digital (49, 69, 656)-net over F9, using
- net defined by OOA [i] based on linear OOA(969, 656, F9, 20, 20) (dual of [(656, 20), 13051, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(969, 6560, F9, 20) (dual of [6560, 6491, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(969, 6560, F9, 20) (dual of [6560, 6491, 21]-code), using
- net defined by OOA [i] based on linear OOA(969, 656, F9, 20, 20) (dual of [(656, 20), 13051, 21]-NRT-code), using
(70−20, 70, 4285)-Net over F9 — Digital
Digital (50, 70, 4285)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(970, 4285, F9, 20) (dual of [4285, 4215, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(970, 6566, F9, 20) (dual of [6566, 6496, 21]-code), using
- 1 times code embedding in larger space [i] based on linear OA(969, 6565, F9, 20) (dual of [6565, 6496, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(969, 6561, F9, 20) (dual of [6561, 6492, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(965, 6561, F9, 19) (dual of [6561, 6496, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 94−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(90, 4, F9, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(90, s, F9, 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(969, 6565, F9, 20) (dual of [6565, 6496, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(970, 6566, F9, 20) (dual of [6566, 6496, 21]-code), using
(70−20, 70, 2707590)-Net in Base 9 — Upper bound on s
There is no (50, 70, 2707591)-net in base 9, because
- the generalized Rao bound for nets shows that 9m ≥ 6 265802 163447 845097 293971 261065 436397 398184 365781 406721 091484 882225 > 970 [i]