Best Known (76−34, 76, s)-Nets in Base 81
(76−34, 76, 730)-Net over F81 — Constructive and digital
Digital (42, 76, 730)-net over F81, using
- t-expansion [i] based on digital (36, 76, 730)-net over F81, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- the Hermitian function field over F81 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 36 and N(F) ≥ 730, using
- net from sequence [i] based on digital (36, 729)-sequence over F81, using
(76−34, 76, 4736)-Net over F81 — Digital
Digital (42, 76, 4736)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8176, 4736, F81, 34) (dual of [4736, 4660, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(8176, 6590, F81, 34) (dual of [6590, 6514, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(23) [i] based on
- linear OA(8167, 6561, F81, 34) (dual of [6561, 6494, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(819, 29, F81, 9) (dual of [29, 20, 10]-code or 29-arc in PG(8,81)), using
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- Reed–Solomon code RS(72,81) [i]
- discarding factors / shortening the dual code based on linear OA(819, 81, F81, 9) (dual of [81, 72, 10]-code or 81-arc in PG(8,81)), using
- construction X applied to Ce(33) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(8176, 6590, F81, 34) (dual of [6590, 6514, 35]-code), using
(76−34, 76, large)-Net in Base 81 — Upper bound on s
There is no (42, 76, large)-net in base 81, because
- 32 times m-reduction [i] would yield (42, 44, large)-net in base 81, but