Best Known (40, 40+37, s)-Nets in Base 81
(40, 40+37, 730)-Net over F81 — Constructive and digital
Digital (40, 77, 730)-net over F81, using
- t-expansion [i] based on digital (36, 77, 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
(40, 40+37, 2727)-Net over F81 — Digital
Digital (40, 77, 2727)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8177, 2727, F81, 2, 37) (dual of [(2727, 2), 5377, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8177, 3287, F81, 2, 37) (dual of [(3287, 2), 6497, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8177, 6574, F81, 37) (dual of [6574, 6497, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(8177, 6575, F81, 37) (dual of [6575, 6498, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- linear OA(8173, 6561, F81, 37) (dual of [6561, 6488, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(8163, 6561, F81, 32) (dual of [6561, 6498, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(36) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(8177, 6575, F81, 37) (dual of [6575, 6498, 38]-code), using
- OOA 2-folding [i] based on linear OA(8177, 6574, F81, 37) (dual of [6574, 6497, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(8177, 3287, F81, 2, 37) (dual of [(3287, 2), 6497, 38]-NRT-code), using
(40, 40+37, large)-Net in Base 81 — Upper bound on s
There is no (40, 77, large)-net in base 81, because
- 35 times m-reduction [i] would yield (40, 42, large)-net in base 81, but