Best Known (51, 51+28, s)-Nets in Base 27
(51, 51+28, 1405)-Net over F27 — Constructive and digital
Digital (51, 79, 1405)-net over F27, using
- net defined by OOA [i] based on linear OOA(2779, 1405, F27, 28, 28) (dual of [(1405, 28), 39261, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(2779, 19670, F27, 28) (dual of [19670, 19591, 29]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(2779, 19670, F27, 28) (dual of [19670, 19591, 29]-code), using
(51, 51+28, 9841)-Net over F27 — Digital
Digital (51, 79, 9841)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2779, 9841, F27, 2, 28) (dual of [(9841, 2), 19603, 29]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2779, 19682, F27, 28) (dual of [19682, 19603, 29]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2779, 19683, F27, 28) (dual of [19683, 19604, 29]-code), using
- OOA 2-folding [i] based on linear OA(2779, 19682, F27, 28) (dual of [19682, 19603, 29]-code), using
(51, 51+28, large)-Net in Base 27 — Upper bound on s
There is no (51, 79, large)-net in base 27, because
- 26 times m-reduction [i] would yield (51, 53, large)-net in base 27, but