Best Known (37−12, 37, s)-Nets in Base 27
(37−12, 37, 3283)-Net over F27 — Constructive and digital
Digital (25, 37, 3283)-net over F27, using
- net defined by OOA [i] based on linear OOA(2737, 3283, F27, 12, 12) (dual of [(3283, 12), 39359, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2737, 19698, F27, 12) (dual of [19698, 19661, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(2734, 19683, F27, 12) (dual of [19683, 19649, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(273, 15, F27, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,27) or 15-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- OA 6-folding and stacking [i] based on linear OA(2737, 19698, F27, 12) (dual of [19698, 19661, 13]-code), using
(37−12, 37, 19698)-Net over F27 — Digital
Digital (25, 37, 19698)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2737, 19698, F27, 12) (dual of [19698, 19661, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(2734, 19683, F27, 12) (dual of [19683, 19649, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [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(273, 15, F27, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,27) or 15-cap in PG(2,27)), using
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- Reed–Solomon code RS(24,27) [i]
- discarding factors / shortening the dual code based on linear OA(273, 27, F27, 3) (dual of [27, 24, 4]-code or 27-arc in PG(2,27) or 27-cap in PG(2,27)), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
(37−12, 37, large)-Net in Base 27 — Upper bound on s
There is no (25, 37, large)-net in base 27, because
- 10 times m-reduction [i] would yield (25, 27, large)-net in base 27, but