Best Known (28−12, 28, s)-Nets in Base 81
(28−12, 28, 1096)-Net over F81 — Constructive and digital
Digital (16, 28, 1096)-net over F81, using
- net defined by OOA [i] based on linear OOA(8128, 1096, F81, 12, 12) (dual of [(1096, 12), 13124, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(8128, 6576, F81, 12) (dual of [6576, 6548, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(8128, 6578, F81, 12) (dual of [6578, 6550, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(8128, 6578, F81, 12) (dual of [6578, 6550, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(8128, 6576, F81, 12) (dual of [6576, 6548, 13]-code), using
(28−12, 28, 6578)-Net over F81 — Digital
Digital (16, 28, 6578)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8128, 6578, F81, 12) (dual of [6578, 6550, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
- linear OA(8123, 6561, F81, 12) (dual of [6561, 6538, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(8111, 6561, F81, 6) (dual of [6561, 6550, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(815, 17, F81, 5) (dual of [17, 12, 6]-code or 17-arc in PG(4,81)), using
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- Reed–Solomon code RS(76,81) [i]
- discarding factors / shortening the dual code based on linear OA(815, 81, F81, 5) (dual of [81, 76, 6]-code or 81-arc in PG(4,81)), using
- construction X applied to Ce(11) ⊂ Ce(5) [i] based on
(28−12, 28, large)-Net in Base 81 — Upper bound on s
There is no (16, 28, large)-net in base 81, because
- 10 times m-reduction [i] would yield (16, 18, large)-net in base 81, but