Best Known (40, 40+21, s)-Nets in Base 27
(40, 40+21, 1968)-Net over F27 — Constructive and digital
Digital (40, 61, 1968)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1968, F27, 21, 21) (dual of [(1968, 21), 41267, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2761, 19681, F27, 21) (dual of [19681, 19620, 22]-code), using
(40, 40+21, 10091)-Net over F27 — Digital
Digital (40, 61, 10091)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2761, 10091, F27, 21) (dual of [10091, 10030, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using
(40, 40+21, large)-Net in Base 27 — Upper bound on s
There is no (40, 61, large)-net in base 27, because
- 19 times m-reduction [i] would yield (40, 42, large)-net in base 27, but