Best Known (48−22, 48, s)-Nets in Base 81
(48−22, 48, 598)-Net over F81 — Constructive and digital
Digital (26, 48, 598)-net over F81, using
- net defined by OOA [i] based on linear OOA(8148, 598, F81, 22, 22) (dual of [(598, 22), 13108, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(8148, 6578, F81, 22) (dual of [6578, 6530, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [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(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(21) ⊂ Ce(15) [i] based on
- OA 11-folding and stacking [i] based on linear OA(8148, 6578, F81, 22) (dual of [6578, 6530, 23]-code), using
(48−22, 48, 3289)-Net over F81 — Digital
Digital (26, 48, 3289)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8148, 3289, F81, 2, 22) (dual of [(3289, 2), 6530, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8148, 6578, F81, 22) (dual of [6578, 6530, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [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(8131, 6561, F81, 16) (dual of [6561, 6530, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(21) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(8148, 6578, F81, 22) (dual of [6578, 6530, 23]-code), using
(48−22, 48, large)-Net in Base 81 — Upper bound on s
There is no (26, 48, large)-net in base 81, because
- 20 times m-reduction [i] would yield (26, 28, large)-net in base 81, but