Best Known (67, 67+18, s)-Nets in Base 9
(67, 67+18, 6563)-Net over F9 — Constructive and digital
Digital (67, 85, 6563)-net over F9, using
- t-expansion [i] based on digital (66, 85, 6563)-net over F9, using
- net defined by OOA [i] based on linear OOA(985, 6563, F9, 19, 19) (dual of [(6563, 19), 124612, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(985, 59068, F9, 19) (dual of [59068, 58983, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(966, 59049, F9, 15) (dual of [59049, 58983, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(94, 19, F9, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,9)), using
- construction X applied to Ce(18) ⊂ Ce(14) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(985, 59068, F9, 19) (dual of [59068, 58983, 20]-code), using
- net defined by OOA [i] based on linear OOA(985, 6563, F9, 19, 19) (dual of [(6563, 19), 124612, 20]-NRT-code), using
(67, 67+18, 59073)-Net over F9 — Digital
Digital (67, 85, 59073)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(985, 59073, F9, 18) (dual of [59073, 58988, 19]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(981, 59049, F9, 19) (dual of [59049, 58968, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(961, 59049, F9, 14) (dual of [59049, 58988, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 59048 = 95−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(94, 24, F9, 3) (dual of [24, 20, 4]-code or 24-cap in PG(3,9)), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
(67, 67+18, large)-Net in Base 9 — Upper bound on s
There is no (67, 85, large)-net in base 9, because
- 16 times m-reduction [i] would yield (67, 69, large)-net in base 9, but