Best Known (57−16, 57, s)-Nets in Base 27
(57−16, 57, 2512)-Net over F27 — Constructive and digital
Digital (41, 57, 2512)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (3, 11, 52)-net over F27, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F27, using
- digital (30, 46, 2460)-net over F27, using
- net defined by OOA [i] based on linear OOA(2746, 2460, F27, 16, 16) (dual of [(2460, 16), 39314, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2746, 19680, F27, 16) (dual of [19680, 19634, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2746, 19680, F27, 16) (dual of [19680, 19634, 17]-code), using
- net defined by OOA [i] based on linear OOA(2746, 2460, F27, 16, 16) (dual of [(2460, 16), 39314, 17]-NRT-code), using
- digital (3, 11, 52)-net over F27, using
(57−16, 57, 2542)-Net in Base 27 — Constructive
(41, 57, 2542)-net in base 27, using
- (u, u+v)-construction [i] based on
- (3, 11, 82)-net in base 27, using
- 1 times m-reduction [i] based on (3, 12, 82)-net in base 27, using
- base change [i] based on digital (0, 9, 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, 9, 82)-net over F81, using
- 1 times m-reduction [i] based on (3, 12, 82)-net in base 27, using
- digital (30, 46, 2460)-net over F27, using
- net defined by OOA [i] based on linear OOA(2746, 2460, F27, 16, 16) (dual of [(2460, 16), 39314, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(2746, 19680, F27, 16) (dual of [19680, 19634, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(2746, 19680, F27, 16) (dual of [19680, 19634, 17]-code), using
- net defined by OOA [i] based on linear OOA(2746, 2460, F27, 16, 16) (dual of [(2460, 16), 39314, 17]-NRT-code), using
- (3, 11, 82)-net in base 27, using
(57−16, 57, 67924)-Net over F27 — Digital
Digital (41, 57, 67924)-net over F27, using
(57−16, 57, large)-Net in Base 27 — Upper bound on s
There is no (41, 57, large)-net in base 27, because
- 14 times m-reduction [i] would yield (41, 43, large)-net in base 27, but