Best Known (75−21, 75, s)-Nets in Base 27
(75−21, 75, 2032)-Net over F27 — Constructive and digital
Digital (54, 75, 2032)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (4, 14, 64)-net over F27, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 4 and N(F) ≥ 64, using
- net from sequence [i] based on digital (4, 63)-sequence over F27, using
- digital (40, 61, 1968)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- digital (4, 14, 64)-net over F27, using
(75−21, 75, 2050)-Net in Base 27 — Constructive
(54, 75, 2050)-net in base 27, using
- (u, u+v)-construction [i] based on
- (4, 14, 82)-net in base 27, using
- 2 times m-reduction [i] based on (4, 16, 82)-net in base 27, using
- base change [i] based on digital (0, 12, 82)-net over F81, using
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F81 with g(F) = 0 and N(F) ≥ 82, using
- the rational function field F81(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 81)-sequence over F81, using
- base change [i] based on digital (0, 12, 82)-net over F81, using
- 2 times m-reduction [i] based on (4, 16, 82)-net in base 27, using
- digital (40, 61, 1968)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- (4, 14, 82)-net in base 27, using
(75−21, 75, 74474)-Net over F27 — Digital
Digital (54, 75, 74474)-net over F27, using
(75−21, 75, large)-Net in Base 27 — Upper bound on s
There is no (54, 75, large)-net in base 27, because
- 19 times m-reduction [i] would yield (54, 56, large)-net in base 27, but