Best Known (68, 68+19, s)-Nets in Base 9
(68, 68+19, 6563)-Net over F9 — Constructive and digital
Digital (68, 87, 6563)-net over F9, using
- 92 times duplication [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
(68, 68+19, 59075)-Net over F9 — Digital
Digital (68, 87, 59075)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(987, 59075, F9, 19) (dual of [59075, 58988, 20]-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(96, 26, F9, 4) (dual of [26, 20, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- 1 times truncation [i] based on linear OA(97, 73, F9, 5) (dual of [73, 66, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(96, 72, F9, 4) (dual of [72, 66, 5]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
(68, 68+19, large)-Net in Base 9 — Upper bound on s
There is no (68, 87, large)-net in base 9, because
- 17 times m-reduction [i] would yield (68, 70, large)-net in base 9, but