Best Known (47−22, 47, s)-Nets in Base 81
(47−22, 47, 597)-Net over F81 — Constructive and digital
Digital (25, 47, 597)-net over F81, using
- t-expansion [i] based on digital (24, 47, 597)-net over F81, using
- net defined by OOA [i] based on linear OOA(8147, 597, F81, 23, 23) (dual of [(597, 23), 13684, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8147, 6568, F81, 23) (dual of [6568, 6521, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(8147, 6569, F81, 23) (dual of [6569, 6522, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(812, 8, F81, 2) (dual of [8, 6, 3]-code or 8-arc in PG(1,81)), using
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- Reed–Solomon code RS(79,81) [i]
- discarding factors / shortening the dual code based on linear OA(812, 81, F81, 2) (dual of [81, 79, 3]-code or 81-arc in PG(1,81)), using
- construction X applied to Ce(22) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(8147, 6569, F81, 23) (dual of [6569, 6522, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8147, 6568, F81, 23) (dual of [6568, 6521, 24]-code), using
- net defined by OOA [i] based on linear OOA(8147, 597, F81, 23, 23) (dual of [(597, 23), 13684, 24]-NRT-code), using
(47−22, 47, 3274)-Net over F81 — Digital
Digital (25, 47, 3274)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8147, 3274, F81, 2, 22) (dual of [(3274, 2), 6501, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8147, 3287, F81, 2, 22) (dual of [(3287, 2), 6527, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8147, 6574, F81, 22) (dual of [6574, 6527, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(8147, 6575, F81, 22) (dual of [6575, 6528, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(8143, 6561, F81, 22) (dual of [6561, 6518, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8147, 6575, F81, 22) (dual of [6575, 6528, 23]-code), using
- OOA 2-folding [i] based on linear OA(8147, 6574, F81, 22) (dual of [6574, 6527, 23]-code), using
- discarding factors / shortening the dual code based on linear OOA(8147, 3287, F81, 2, 22) (dual of [(3287, 2), 6527, 23]-NRT-code), using
(47−22, 47, large)-Net in Base 81 — Upper bound on s
There is no (25, 47, large)-net in base 81, because
- 20 times m-reduction [i] would yield (25, 27, large)-net in base 81, but