Best Known (40, 75, s)-Nets in Base 81
(40, 75, 730)-Net over F81 — Constructive and digital
Digital (40, 75, 730)-net over F81, using
- t-expansion [i] based on digital (36, 75, 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, 75, 3290)-Net over F81 — Digital
Digital (40, 75, 3290)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8175, 3290, F81, 2, 35) (dual of [(3290, 2), 6505, 36]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8175, 6580, F81, 35) (dual of [6580, 6505, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(8175, 6581, F81, 35) (dual of [6581, 6506, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(27) [i] based on
- linear OA(8169, 6561, F81, 35) (dual of [6561, 6492, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(34) ⊂ Ce(27) [i] based on
- discarding factors / shortening the dual code based on linear OA(8175, 6581, F81, 35) (dual of [6581, 6506, 36]-code), using
- OOA 2-folding [i] based on linear OA(8175, 6580, F81, 35) (dual of [6580, 6505, 36]-code), using
(40, 75, large)-Net in Base 81 — Upper bound on s
There is no (40, 75, large)-net in base 81, because
- 33 times m-reduction [i] would yield (40, 42, large)-net in base 81, but