Best Known (47, 47+24, s)-Nets in Base 27
(47, 47+24, 1640)-Net over F27 — Constructive and digital
Digital (47, 71, 1640)-net over F27, using
- 271 times duplication [i] based on digital (46, 70, 1640)-net over F27, using
- net defined by OOA [i] based on linear OOA(2770, 1640, F27, 24, 24) (dual of [(1640, 24), 39290, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2770, 19680, F27, 24) (dual of [19680, 19610, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2770, 19680, F27, 24) (dual of [19680, 19610, 25]-code), using
- net defined by OOA [i] based on linear OOA(2770, 1640, F27, 24, 24) (dual of [(1640, 24), 39290, 25]-NRT-code), using
(47, 47+24, 12469)-Net over F27 — Digital
Digital (47, 71, 12469)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2771, 12469, F27, 24) (dual of [12469, 12398, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2771, 19690, F27, 24) (dual of [19690, 19619, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2764, 19683, F27, 22) (dual of [19683, 19619, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(23) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(2771, 19690, F27, 24) (dual of [19690, 19619, 25]-code), using
(47, 47+24, large)-Net in Base 27 — Upper bound on s
There is no (47, 71, large)-net in base 27, because
- 22 times m-reduction [i] would yield (47, 49, large)-net in base 27, but