Best Known (33, 33+28, s)-Nets in Base 81
(33, 33+28, 486)-Net over F81 — Constructive and digital
Digital (33, 61, 486)-net over F81, using
- 2 times m-reduction [i] based on digital (33, 63, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 17, 116)-net over F81, using
- net from sequence [i] based on digital (2, 115)-sequence over F81, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 2 and N(F) ≥ 116, using
- net from sequence [i] based on digital (2, 115)-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 (2, 17, 116)-net over F81, using
- (u, u+v)-construction [i] based on
(33, 33+28, 3333)-Net over F81 — Digital
Digital (33, 61, 3333)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8161, 3333, F81, 28) (dual of [3333, 3272, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, 6581, F81, 28) (dual of [6581, 6520, 29]-code), using
- construction X applied to Ce(27) ⊂ Ce(20) [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(8141, 6561, F81, 21) (dual of [6561, 6520, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(27) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(8161, 6581, F81, 28) (dual of [6581, 6520, 29]-code), using
(33, 33+28, large)-Net in Base 81 — Upper bound on s
There is no (33, 61, large)-net in base 81, because
- 26 times m-reduction [i] would yield (33, 35, large)-net in base 81, but