Best Known (30, 30+26, s)-Nets in Base 81
(30, 30+26, 506)-Net over F81 — Constructive and digital
Digital (30, 56, 506)-net over F81, using
- net defined by OOA [i] based on linear OOA(8156, 506, F81, 26, 26) (dual of [(506, 26), 13100, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(8156, 6578, F81, 26) (dual of [6578, 6522, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(25) ⊂ Ce(19) [i] based on
- OA 13-folding and stacking [i] based on linear OA(8156, 6578, F81, 26) (dual of [6578, 6522, 27]-code), using
(30, 30+26, 3289)-Net over F81 — Digital
Digital (30, 56, 3289)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8156, 3289, F81, 2, 26) (dual of [(3289, 2), 6522, 27]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8156, 6578, F81, 26) (dual of [6578, 6522, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8139, 6561, F81, 20) (dual of [6561, 6522, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(25) ⊂ Ce(19) [i] based on
- OOA 2-folding [i] based on linear OA(8156, 6578, F81, 26) (dual of [6578, 6522, 27]-code), using
(30, 30+26, large)-Net in Base 81 — Upper bound on s
There is no (30, 56, large)-net in base 81, because
- 24 times m-reduction [i] would yield (30, 32, large)-net in base 81, but