Best Known (61−27, 61, s)-Nets in Base 81
(61−27, 61, 530)-Net over F81 — Constructive and digital
Digital (34, 61, 530)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 160)-net over F81, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 5 and N(F) ≥ 160, using
- net from sequence [i] based on digital (5, 159)-sequence over F81, using
- digital (16, 43, 370)-net over F81, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 16 and N(F) ≥ 370, using
- net from sequence [i] based on digital (16, 369)-sequence over F81, using
- digital (5, 18, 160)-net over F81, using
(61−27, 61, 4829)-Net over F81 — Digital
Digital (34, 61, 4829)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8161, 4829, F81, 27) (dual of [4829, 4768, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, 6587, F81, 27) (dual of [6587, 6526, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(17) [i] based on
- linear OA(8153, 6561, F81, 27) (dual of [6561, 6508, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8135, 6561, F81, 18) (dual of [6561, 6526, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(818, 26, F81, 8) (dual of [26, 18, 9]-code or 26-arc in PG(7,81)), using
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- Reed–Solomon code RS(73,81) [i]
- discarding factors / shortening the dual code based on linear OA(818, 81, F81, 8) (dual of [81, 73, 9]-code or 81-arc in PG(7,81)), using
- construction X applied to Ce(26) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(8161, 6587, F81, 27) (dual of [6587, 6526, 28]-code), using
(61−27, 61, large)-Net in Base 81 — Upper bound on s
There is no (34, 61, large)-net in base 81, because
- 25 times m-reduction [i] would yield (34, 36, large)-net in base 81, but