Best Known (41, 41+36, s)-Nets in Base 81
(41, 41+36, 730)-Net over F81 — Constructive and digital
Digital (41, 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
(41, 41+36, 3290)-Net over F81 — Digital
Digital (41, 77, 3290)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8177, 3290, F81, 2, 36) (dual of [(3290, 2), 6503, 37]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8177, 6580, F81, 36) (dual of [6580, 6503, 37]-code), using
- discarding factors / shortening the dual code based on linear OA(8177, 6581, F81, 36) (dual of [6581, 6504, 37]-code), using
- construction X applied to Ce(35) ⊂ Ce(28) [i] based on
- linear OA(8171, 6561, F81, 36) (dual of [6561, 6490, 37]-code), using an extension Ce(35) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,35], and designed minimum distance d ≥ |I|+1 = 36 [i]
- linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(816, 20, F81, 6) (dual of [20, 14, 7]-code or 20-arc in PG(5,81)), using
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- Reed–Solomon code RS(75,81) [i]
- discarding factors / shortening the dual code based on linear OA(816, 81, F81, 6) (dual of [81, 75, 7]-code or 81-arc in PG(5,81)), using
- construction X applied to Ce(35) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(8177, 6581, F81, 36) (dual of [6581, 6504, 37]-code), using
- OOA 2-folding [i] based on linear OA(8177, 6580, F81, 36) (dual of [6580, 6503, 37]-code), using
(41, 41+36, large)-Net in Base 81 — Upper bound on s
There is no (41, 77, large)-net in base 81, because
- 34 times m-reduction [i] would yield (41, 43, large)-net in base 81, but