Best Known (69, 69+27, s)-Nets in Base 27
(69, 69+27, 1581)-Net over F27 — Constructive and digital
Digital (69, 96, 1581)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (5, 18, 68)-net over F27, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 5 and N(F) ≥ 68, using
- net from sequence [i] based on digital (5, 67)-sequence over F27, using
- digital (51, 78, 1513)-net over F27, using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- digital (5, 18, 68)-net over F27, using
(69, 69+27, 1595)-Net in Base 27 — Constructive
(69, 96, 1595)-net in base 27, using
- (u, u+v)-construction [i] based on
- (5, 18, 82)-net in base 27, using
- 2 times m-reduction [i] based on (5, 20, 82)-net in base 27, using
- base change [i] based on digital (0, 15, 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, 15, 82)-net over F81, using
- 2 times m-reduction [i] based on (5, 20, 82)-net in base 27, using
- digital (51, 78, 1513)-net over F27, using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2778, 19682, F27, 27) (dual of [19682, 19604, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2778, 19670, F27, 27) (dual of [19670, 19592, 28]-code), using
- net defined by OOA [i] based on linear OOA(2778, 1513, F27, 27, 27) (dual of [(1513, 27), 40773, 28]-NRT-code), using
- (5, 18, 82)-net in base 27, using
(69, 69+27, 78240)-Net over F27 — Digital
Digital (69, 96, 78240)-net over F27, using
(69, 69+27, large)-Net in Base 27 — Upper bound on s
There is no (69, 96, large)-net in base 27, because
- 25 times m-reduction [i] would yield (69, 71, large)-net in base 27, but