Best Known (68, 68+13, s)-Nets in Base 3
(68, 68+13, 9841)-Net over F3 — Constructive and digital
Digital (68, 81, 9841)-net over F3, using
- net defined by OOA [i] based on linear OOA(381, 9841, F3, 13, 13) (dual of [(9841, 13), 127852, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(381, 59047, F3, 13) (dual of [59047, 58966, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(381, 59047, F3, 13) (dual of [59047, 58966, 14]-code), using
(68, 68+13, 19683)-Net over F3 — Digital
Digital (68, 81, 19683)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(381, 19683, F3, 3, 13) (dual of [(19683, 3), 58968, 14]-NRT-code), using
- OOA 3-folding [i] based on linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 59048 = 310−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- OOA 3-folding [i] based on linear OA(381, 59049, F3, 13) (dual of [59049, 58968, 14]-code), using
(68, 68+13, 3441978)-Net in Base 3 — Upper bound on s
There is no (68, 81, 3441979)-net in base 3, because
- 1 times m-reduction [i] would yield (68, 80, 3441979)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 147 809020 557944 057828 536579 265307 689037 > 380 [i]