Best Known (26, 46, s)-Nets in Base 81
(26, 46, 658)-Net over F81 — Constructive and digital
Digital (26, 46, 658)-net over F81, using
- 1 times m-reduction [i] based on digital (26, 47, 658)-net over F81, using
- net defined by OOA [i] based on linear OOA(8147, 658, F81, 21, 21) (dual of [(658, 21), 13771, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8147, 6581, F81, 21) (dual of [6581, 6534, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(13) [i] based on
- linear OA(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(8127, 6561, F81, 14) (dual of [6561, 6534, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-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(20) ⊂ Ce(13) [i] based on
- OOA 10-folding and stacking with additional row [i] based on linear OA(8147, 6581, F81, 21) (dual of [6581, 6534, 22]-code), using
- net defined by OOA [i] based on linear OOA(8147, 658, F81, 21, 21) (dual of [(658, 21), 13771, 22]-NRT-code), using
(26, 46, 5567)-Net over F81 — Digital
Digital (26, 46, 5567)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8146, 5567, F81, 20) (dual of [5567, 5521, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8146, 6584, F81, 20) (dual of [6584, 6538, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
- 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(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(817, 23, F81, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to Ce(19) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(8146, 6584, F81, 20) (dual of [6584, 6538, 21]-code), using
(26, 46, large)-Net in Base 81 — Upper bound on s
There is no (26, 46, large)-net in base 81, because
- 18 times m-reduction [i] would yield (26, 28, large)-net in base 81, but