Best Known (25−9, 25, s)-Nets in Base 27
(25−9, 25, 4921)-Net over F27 — Constructive and digital
Digital (16, 25, 4921)-net over F27, using
- net defined by OOA [i] based on linear OOA(2725, 4921, F27, 9, 9) (dual of [(4921, 9), 44264, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2725, 19685, F27, 9) (dual of [19685, 19660, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2725, 19686, F27, 9) (dual of [19686, 19661, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(2725, 19683, F27, 9) (dual of [19683, 19658, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2722, 19683, F27, 8) (dual of [19683, 19661, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- discarding factors / shortening the dual code based on linear OA(2725, 19686, F27, 9) (dual of [19686, 19661, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2725, 19685, F27, 9) (dual of [19685, 19660, 10]-code), using
(25−9, 25, 10504)-Net over F27 — Digital
Digital (16, 25, 10504)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2725, 10504, F27, 9) (dual of [10504, 10479, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2725, 19683, F27, 9) (dual of [19683, 19658, 10]-code), using
- an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- discarding factors / shortening the dual code based on linear OA(2725, 19683, F27, 9) (dual of [19683, 19658, 10]-code), using
(25−9, 25, large)-Net in Base 27 — Upper bound on s
There is no (16, 25, large)-net in base 27, because
- 7 times m-reduction [i] would yield (16, 18, large)-net in base 27, but