Best Known (58−20, 58, s)-Nets in Base 27
(58−20, 58, 1968)-Net over F27 — Constructive and digital
Digital (38, 58, 1968)-net over F27, using
- net defined by OOA [i] based on linear OOA(2758, 1968, F27, 20, 20) (dual of [(1968, 20), 39302, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(2758, 19680, F27, 20) (dual of [19680, 19622, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(2758, 19680, F27, 20) (dual of [19680, 19622, 21]-code), using
(58−20, 58, 9895)-Net over F27 — Digital
Digital (38, 58, 9895)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2758, 9895, F27, 20) (dual of [9895, 9837, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using
- an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- discarding factors / shortening the dual code based on linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using
(58−20, 58, large)-Net in Base 27 — Upper bound on s
There is no (38, 58, large)-net in base 27, because
- 18 times m-reduction [i] would yield (38, 40, large)-net in base 27, but