Best Known (35, 35+28, s)-Nets in Base 81
(35, 35+28, 530)-Net over F81 — Constructive and digital
Digital (35, 63, 530)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (5, 19, 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, 44, 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, 19, 160)-net over F81, using
(35, 35+28, 4678)-Net over F81 — Digital
Digital (35, 63, 4678)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8163, 4678, F81, 28) (dual of [4678, 4615, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 6587, F81, 28) (dual of [6587, 6524, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(18) [i] based on
- linear OA(8155, 6561, F81, 28) (dual of [6561, 6506, 29]-code), using an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [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(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(27) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(8163, 6587, F81, 28) (dual of [6587, 6524, 29]-code), using
(35, 35+28, large)-Net in Base 81 — Upper bound on s
There is no (35, 63, large)-net in base 81, because
- 26 times m-reduction [i] would yield (35, 37, large)-net in base 81, but