Best Known (85−19, 85, s)-Nets in Base 9
(85−19, 85, 6563)-Net over F9 — Constructive and digital
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
(85−19, 85, 46542)-Net over F9 — Digital
Digital (66, 85, 46542)-net over F9, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(985, 46542, F9, 19) (dual of [46542, 46457, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(985, 59064, F9, 19) (dual of [59064, 58979, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- linear OA(981, 59050, F9, 19) (dual of [59050, 58969, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(971, 59050, F9, 15) (dual of [59050, 58979, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 59050 | 910−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- linear OA(94, 14, F9, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,9)), using
- construction X applied to C([0,9]) ⊂ C([0,7]) [i] based on
- discarding factors / shortening the dual code based on linear OA(985, 59064, F9, 19) (dual of [59064, 58979, 20]-code), using
(85−19, 85, large)-Net in Base 9 — Upper bound on s
There is no (66, 85, large)-net in base 9, because
- 17 times m-reduction [i] would yield (66, 68, large)-net in base 9, but