Best Known (36, 66, s)-Nets in Base 81
(36, 66, 730)-Net over F81 — Constructive and digital
Digital (36, 66, 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
(36, 66, 3792)-Net over F81 — Digital
Digital (36, 66, 3792)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8166, 3792, F81, 30) (dual of [3792, 3726, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8166, 6584, F81, 30) (dual of [6584, 6518, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(21) [i] based on
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- 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(817, 23, F81, 7) (dual of [23, 16, 8]-code or 23-arc in PG(6,81)), using
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- Reed–Solomon code RS(74,81) [i]
- discarding factors / shortening the dual code based on linear OA(817, 81, F81, 7) (dual of [81, 74, 8]-code or 81-arc in PG(6,81)), using
- construction X applied to Ce(29) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8166, 6584, F81, 30) (dual of [6584, 6518, 31]-code), using
(36, 66, large)-Net in Base 81 — Upper bound on s
There is no (36, 66, large)-net in base 81, because
- 28 times m-reduction [i] would yield (36, 38, large)-net in base 81, but