Best Known (65−30, 65, s)-Nets in Base 81
(65−30, 65, 520)-Net over F81 — Constructive and digital
Digital (35, 65, 520)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (4, 19, 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, 46, 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, 19, 150)-net over F81, using
(65−30, 65, 3290)-Net over F81 — Digital
Digital (35, 65, 3290)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8165, 3290, F81, 2, 30) (dual of [(3290, 2), 6515, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8165, 6580, F81, 30) (dual of [6580, 6515, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, 6581, F81, 30) (dual of [6581, 6516, 31]-code), using
- construction X applied to Ce(29) ⊂ Ce(22) [i] based on
- linear OA(8159, 6561, F81, 30) (dual of [6561, 6502, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [i]
- linear OA(8145, 6561, F81, 23) (dual of [6561, 6516, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [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(29) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8165, 6581, F81, 30) (dual of [6581, 6516, 31]-code), using
- OOA 2-folding [i] based on linear OA(8165, 6580, F81, 30) (dual of [6580, 6515, 31]-code), using
(65−30, 65, large)-Net in Base 81 — Upper bound on s
There is no (35, 65, large)-net in base 81, because
- 28 times m-reduction [i] would yield (35, 37, large)-net in base 81, but