Best Known (75−32, 75, s)-Nets in Base 81
(75−32, 75, 730)-Net over F81 — Constructive and digital
Digital (43, 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
(75−32, 75, 6599)-Net over F81 — Digital
Digital (43, 75, 6599)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8175, 6599, F81, 32) (dual of [6599, 6524, 33]-code), using
- construction X applied to Ce(31) ⊂ Ce(18) [i] based on
- 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(8137, 6561, F81, 19) (dual of [6561, 6524, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(8112, 38, F81, 12) (dual of [38, 26, 13]-code or 38-arc in PG(11,81)), using
- discarding factors / shortening the dual code based on linear OA(8112, 81, F81, 12) (dual of [81, 69, 13]-code or 81-arc in PG(11,81)), using
- Reed–Solomon code RS(69,81) [i]
- discarding factors / shortening the dual code based on linear OA(8112, 81, F81, 12) (dual of [81, 69, 13]-code or 81-arc in PG(11,81)), using
- construction X applied to Ce(31) ⊂ Ce(18) [i] based on
(75−32, 75, large)-Net in Base 81 — Upper bound on s
There is no (43, 75, large)-net in base 81, because
- 30 times m-reduction [i] would yield (43, 45, large)-net in base 81, but