Best Known (22, 22+10, s)-Nets in Base 27
(22, 22+10, 3940)-Net over F27 — Constructive and digital
Digital (22, 32, 3940)-net over F27, using
- net defined by OOA [i] based on linear OOA(2732, 3940, F27, 10, 10) (dual of [(3940, 10), 39368, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2732, 19700, F27, 10) (dual of [19700, 19668, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2732, 19702, F27, 10) (dual of [19702, 19670, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(2728, 19683, F27, 10) (dual of [19683, 19655, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2713, 19683, F27, 5) (dual of [19683, 19670, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(274, 19, F27, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(2732, 19702, F27, 10) (dual of [19702, 19670, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2732, 19700, F27, 10) (dual of [19700, 19668, 11]-code), using
(22, 22+10, 19702)-Net over F27 — Digital
Digital (22, 32, 19702)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2732, 19702, F27, 10) (dual of [19702, 19670, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
- linear OA(2728, 19683, F27, 10) (dual of [19683, 19655, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(2713, 19683, F27, 5) (dual of [19683, 19670, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(274, 19, F27, 4) (dual of [19, 15, 5]-code or 19-arc in PG(3,27)), using
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- Reed–Solomon code RS(23,27) [i]
- discarding factors / shortening the dual code based on linear OA(274, 27, F27, 4) (dual of [27, 23, 5]-code or 27-arc in PG(3,27)), using
- construction X applied to Ce(9) ⊂ Ce(4) [i] based on
(22, 22+10, large)-Net in Base 27 — Upper bound on s
There is no (22, 32, large)-net in base 27, because
- 8 times m-reduction [i] would yield (22, 24, large)-net in base 27, but