Best Known (59−26, 59, s)-Nets in Base 81
(59−26, 59, 520)-Net over F81 — Constructive and digital
Digital (33, 59, 520)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 17, 150)-net over F81, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 4 and N(F) ≥ 150, using
- net from sequence [i] based on digital (4, 149)-sequence over F81, using
- digital (16, 42, 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 (4, 17, 150)-net over F81, using
(59−26, 59, 5006)-Net over F81 — Digital
Digital (33, 59, 5006)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8159, 5006, F81, 26) (dual of [5006, 4947, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(8159, 6587, F81, 26) (dual of [6587, 6528, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(16) [i] based on
- linear OA(8151, 6561, F81, 26) (dual of [6561, 6510, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(8133, 6561, F81, 17) (dual of [6561, 6528, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(25) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(8159, 6587, F81, 26) (dual of [6587, 6528, 27]-code), using
(59−26, 59, large)-Net in Base 81 — Upper bound on s
There is no (33, 59, large)-net in base 81, because
- 24 times m-reduction [i] would yield (33, 35, large)-net in base 81, but