Best Known (20, 20+16, s)-Nets in Base 81
(20, 20+16, 822)-Net over F81 — Constructive and digital
Digital (20, 36, 822)-net over F81, using
- net defined by OOA [i] based on linear OOA(8136, 822, F81, 16, 16) (dual of [(822, 16), 13116, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8136, 6576, F81, 16) (dual of [6576, 6540, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8136, 6578, F81, 16) (dual of [6578, 6542, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- 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(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(15) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(8136, 6578, F81, 16) (dual of [6578, 6542, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(8136, 6576, F81, 16) (dual of [6576, 6540, 17]-code), using
(20, 20+16, 4456)-Net over F81 — Digital
Digital (20, 36, 4456)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8136, 4456, F81, 16) (dual of [4456, 4420, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8136, 6578, F81, 16) (dual of [6578, 6542, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(9) [i] based on
- 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(8119, 6561, F81, 10) (dual of [6561, 6542, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(15) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(8136, 6578, F81, 16) (dual of [6578, 6542, 17]-code), using
(20, 20+16, large)-Net in Base 81 — Upper bound on s
There is no (20, 36, large)-net in base 81, because
- 14 times m-reduction [i] would yield (20, 22, large)-net in base 81, but