Best Known (42, 42+33, s)-Nets in Base 81
(42, 42+33, 730)-Net over F81 — Constructive and digital
Digital (42, 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
(42, 42+33, 5566)-Net over F81 — Digital
Digital (42, 75, 5566)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8175, 5566, F81, 33) (dual of [5566, 5491, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(8175, 6593, F81, 33) (dual of [6593, 6518, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(21) [i] based on
- linear OA(8165, 6561, F81, 33) (dual of [6561, 6496, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [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(8110, 32, F81, 10) (dual of [32, 22, 11]-code or 32-arc in PG(9,81)), using
- discarding factors / shortening the dual code based on linear OA(8110, 81, F81, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
- Reed–Solomon code RS(71,81) [i]
- discarding factors / shortening the dual code based on linear OA(8110, 81, F81, 10) (dual of [81, 71, 11]-code or 81-arc in PG(9,81)), using
- construction X applied to Ce(32) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(8175, 6593, F81, 33) (dual of [6593, 6518, 34]-code), using
(42, 42+33, large)-Net in Base 81 — Upper bound on s
There is no (42, 75, large)-net in base 81, because
- 31 times m-reduction [i] would yield (42, 44, large)-net in base 81, but