Best Known (49−16, 49, s)-Nets in Base 81
(49−16, 49, 66432)-Net over F81 — Constructive and digital
Digital (33, 49, 66432)-net over F81, using
- net defined by OOA [i] based on linear OOA(8149, 66432, F81, 16, 16) (dual of [(66432, 16), 1062863, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(8149, 531456, F81, 16) (dual of [531456, 531407, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- OA 8-folding and stacking [i] based on linear OA(8149, 531456, F81, 16) (dual of [531456, 531407, 17]-code), using
(49−16, 49, 265728)-Net over F81 — Digital
Digital (33, 49, 265728)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8149, 265728, F81, 2, 16) (dual of [(265728, 2), 531407, 17]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8149, 531456, F81, 16) (dual of [531456, 531407, 17]-code), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- linear OA(8146, 531441, F81, 16) (dual of [531441, 531395, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(8134, 531441, F81, 12) (dual of [531441, 531407, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 813−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(813, 15, F81, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,81) or 15-cap in PG(2,81)), using
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- Reed–Solomon code RS(78,81) [i]
- discarding factors / shortening the dual code based on linear OA(813, 81, F81, 3) (dual of [81, 78, 4]-code or 81-arc in PG(2,81) or 81-cap in PG(2,81)), using
- construction X applied to Ce(15) ⊂ Ce(11) [i] based on
- OOA 2-folding [i] based on linear OA(8149, 531456, F81, 16) (dual of [531456, 531407, 17]-code), using
(49−16, 49, large)-Net in Base 81 — Upper bound on s
There is no (33, 49, large)-net in base 81, because
- 14 times m-reduction [i] would yield (33, 35, large)-net in base 81, but