Best Known (81−20, 81, s)-Nets in Base 9
(81−20, 81, 1313)-Net over F9 — Constructive and digital
Digital (61, 81, 1313)-net over F9, using
- 91 times duplication [i] based on digital (60, 80, 1313)-net over F9, using
- net defined by OOA [i] based on linear OOA(980, 1313, F9, 20, 20) (dual of [(1313, 20), 26180, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(980, 13130, F9, 20) (dual of [13130, 13050, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(980, 13132, F9, 20) (dual of [13132, 13052, 21]-code), using
- trace code [i] based on linear OA(8140, 6566, F81, 20) (dual of [6566, 6526, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(8140, 6566, F81, 20) (dual of [6566, 6526, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(980, 13132, F9, 20) (dual of [13132, 13052, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(980, 13130, F9, 20) (dual of [13130, 13050, 21]-code), using
- net defined by OOA [i] based on linear OOA(980, 1313, F9, 20, 20) (dual of [(1313, 20), 26180, 21]-NRT-code), using
(81−20, 81, 13134)-Net over F9 — Digital
Digital (61, 81, 13134)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(981, 13134, F9, 20) (dual of [13134, 13053, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(980, 13132, F9, 20) (dual of [13132, 13052, 21]-code), using
- trace code [i] based on linear OA(8140, 6566, F81, 20) (dual of [6566, 6526, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(811, 5, F81, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(811, s, F81, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(19) ⊂ Ce(17) [i] based on
- trace code [i] based on linear OA(8140, 6566, F81, 20) (dual of [6566, 6526, 21]-code), using
- linear OA(980, 13133, F9, 19) (dual of [13133, 13053, 20]-code), using Gilbert–Varšamov bound and bm = 980 > Vbs−1(k−1) = 375 186158 805992 945468 489913 144530 796260 714022 396373 276480 841451 218893 023969 [i]
- linear OA(90, 1, F9, 0) (dual of [1, 1, 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
- linear OA(980, 13132, F9, 20) (dual of [13132, 13052, 21]-code), using
- construction X with Varšamov bound [i] based on
(81−20, 81, large)-Net in Base 9 — Upper bound on s
There is no (61, 81, large)-net in base 9, because
- 18 times m-reduction [i] would yield (61, 63, large)-net in base 9, but