Best Known (61−29, 61, s)-Nets in Base 81
(61−29, 61, 486)-Net over F81 — Constructive and digital
Digital (32, 61, 486)-net over F81, using
- (u, u+v)-construction [i] based on
- digital (2, 16, 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, 45, 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, 16, 116)-net over F81, using
(61−29, 61, 2813)-Net over F81 — Digital
Digital (32, 61, 2813)-net over F81, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(8161, 2813, F81, 2, 29) (dual of [(2813, 2), 5565, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(8161, 3287, F81, 2, 29) (dual of [(3287, 2), 6513, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(8161, 6574, F81, 29) (dual of [6574, 6513, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(8161, 6575, F81, 29) (dual of [6575, 6514, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- linear OA(8157, 6561, F81, 29) (dual of [6561, 6504, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(8147, 6561, F81, 24) (dual of [6561, 6514, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(814, 14, F81, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,81)), using
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- Reed–Solomon code RS(77,81) [i]
- discarding factors / shortening the dual code based on linear OA(814, 81, F81, 4) (dual of [81, 77, 5]-code or 81-arc in PG(3,81)), using
- construction X applied to Ce(28) ⊂ Ce(23) [i] based on
- discarding factors / shortening the dual code based on linear OA(8161, 6575, F81, 29) (dual of [6575, 6514, 30]-code), using
- OOA 2-folding [i] based on linear OA(8161, 6574, F81, 29) (dual of [6574, 6513, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(8161, 3287, F81, 2, 29) (dual of [(3287, 2), 6513, 30]-NRT-code), using
(61−29, 61, large)-Net in Base 81 — Upper bound on s
There is no (32, 61, large)-net in base 81, because
- 27 times m-reduction [i] would yield (32, 34, large)-net in base 81, but