Best Known (33−16, 33, s)-Nets in Base 81
(33−16, 33, 821)-Net over F81 — Constructive and digital
Digital (17, 33, 821)-net over F81, using
- net defined by OOA [i] based on linear OOA(8133, 821, F81, 16, 16) (dual of [(821, 16), 13103, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8133, 6568, F81, 16) (dual of [6568, 6535, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, 6569, F81, 16) (dual of [6569, 6536, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(8133, 6569, F81, 16) (dual of [6569, 6536, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(8133, 6568, F81, 16) (dual of [6568, 6535, 17]-code), using
(33−16, 33, 2514)-Net over F81 — Digital
Digital (17, 33, 2514)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8133, 2514, F81, 2, 16) (dual of [(2514, 2), 4995, 17]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8133, 3284, F81, 2, 16) (dual of [(3284, 2), 6535, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8133, 6568, F81, 16) (dual of [6568, 6535, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(8133, 6569, F81, 16) (dual of [6569, 6536, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(12) [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(8125, 6561, F81, 13) (dual of [6561, 6536, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(15) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(8133, 6569, F81, 16) (dual of [6569, 6536, 17]-code), using
- OOA 2-folding [i] based on linear OA(8133, 6568, F81, 16) (dual of [6568, 6535, 17]-code), using
- discarding factors / shortening the dual code based on linear OOA(8133, 3284, F81, 2, 16) (dual of [(3284, 2), 6535, 17]-NRT-code), using
(33−16, 33, 3508329)-Net in Base 81 — Upper bound on s
There is no (17, 33, 3508330)-net in base 81, because
- the generalized Rao bound for nets shows that 81m ≥ 955 006736 128917 582080 999993 292259 510097 681083 723177 042514 259201 > 8133 [i]