Best Known (48, 70, s)-Nets in Base 81
(48, 70, 48315)-Net over F81 — Constructive and digital
Digital (48, 70, 48315)-net over F81, using
- net defined by OOA [i] based on linear OOA(8170, 48315, F81, 22, 22) (dual of [(48315, 22), 1062860, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8170, 531465, F81, 22) (dual of [531465, 531395, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 531468, F81, 22) (dual of [531468, 531398, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 531468, F81, 22) (dual of [531468, 531398, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(8170, 531465, F81, 22) (dual of [531465, 531395, 23]-code), using
(48, 70, 398547)-Net over F81 — Digital
Digital (48, 70, 398547)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8170, 398547, F81, 22) (dual of [398547, 398477, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8170, 531468, F81, 22) (dual of [531468, 531398, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- linear OA(8164, 531441, F81, 22) (dual of [531441, 531377, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8143, 531441, F81, 15) (dual of [531441, 531398, 16]-code), using an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(816, 27, F81, 6) (dual of [27, 21, 7]-code or 27-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(21) ⊂ Ce(14) [i] based on
- discarding factors / shortening the dual code based on linear OA(8170, 531468, F81, 22) (dual of [531468, 531398, 23]-code), using
(48, 70, large)-Net in Base 81 — Upper bound on s
There is no (48, 70, large)-net in base 81, because
- 20 times m-reduction [i] would yield (48, 50, large)-net in base 81, but