Best Known (71, 71+38, s)-Nets in Base 27
(71, 71+38, 1036)-Net over F27 — Constructive and digital
Digital (71, 109, 1036)-net over F27, using
- net defined by OOA [i] based on linear OOA(27109, 1036, F27, 38, 38) (dual of [(1036, 38), 39259, 39]-NRT-code), using
- OA 19-folding and stacking [i] based on linear OA(27109, 19684, F27, 38) (dual of [19684, 19575, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(27109, 19686, F27, 38) (dual of [19686, 19577, 39]-code), using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- linear OA(27109, 19683, F27, 38) (dual of [19683, 19574, 39]-code), using an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(27106, 19683, F27, 37) (dual of [19683, 19577, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(270, 3, F27, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(37) ⊂ Ce(36) [i] based on
- discarding factors / shortening the dual code based on linear OA(27109, 19686, F27, 38) (dual of [19686, 19577, 39]-code), using
- OA 19-folding and stacking [i] based on linear OA(27109, 19684, F27, 38) (dual of [19684, 19575, 39]-code), using
(71, 71+38, 10792)-Net over F27 — Digital
Digital (71, 109, 10792)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27109, 10792, F27, 38) (dual of [10792, 10683, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(27109, 19683, F27, 38) (dual of [19683, 19574, 39]-code), using
- an extension Ce(37) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,37], and designed minimum distance d ≥ |I|+1 = 38 [i]
- discarding factors / shortening the dual code based on linear OA(27109, 19683, F27, 38) (dual of [19683, 19574, 39]-code), using
(71, 71+38, large)-Net in Base 27 — Upper bound on s
There is no (71, 109, large)-net in base 27, because
- 36 times m-reduction [i] would yield (71, 73, large)-net in base 27, but