Best Known (27−9, 27, s)-Nets in Base 27
(27−9, 27, 4923)-Net over F27 — Constructive and digital
Digital (18, 27, 4923)-net over F27, using
- net defined by OOA [i] based on linear OOA(2727, 4923, F27, 9, 9) (dual of [(4923, 9), 44280, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2727, 19693, F27, 9) (dual of [19693, 19666, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(2727, 19694, F27, 9) (dual of [19694, 19667, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [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(2716, 19683, F27, 6) (dual of [19683, 19667, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(2727, 19694, F27, 9) (dual of [19694, 19667, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(2727, 19693, F27, 9) (dual of [19693, 19666, 10]-code), using
(27−9, 27, 19694)-Net over F27 — Digital
Digital (18, 27, 19694)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2727, 19694, F27, 9) (dual of [19694, 19667, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(5) [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(2716, 19683, F27, 6) (dual of [19683, 19667, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(8) ⊂ Ce(5) [i] based on
(27−9, 27, large)-Net in Base 27 — Upper bound on s
There is no (18, 27, large)-net in base 27, because
- 7 times m-reduction [i] would yield (18, 20, large)-net in base 27, but