Best Known (36, 36+12, s)-Nets in Base 27
(36, 36+12, 88576)-Net over F27 — Constructive and digital
Digital (36, 48, 88576)-net over F27, using
- net defined by OOA [i] based on linear OOA(2748, 88576, F27, 12, 12) (dual of [(88576, 12), 1062864, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(2748, 531456, F27, 12) (dual of [531456, 531408, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(2748, 531460, F27, 12) (dual of [531460, 531412, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2729, 531441, F27, 8) (dual of [531441, 531412, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(273, 19, F27, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,27) or 19-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
- discarding factors / shortening the dual code based on linear OA(2748, 531460, F27, 12) (dual of [531460, 531412, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(2748, 531456, F27, 12) (dual of [531456, 531408, 13]-code), using
(36, 36+12, 531460)-Net over F27 — Digital
Digital (36, 48, 531460)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2748, 531460, F27, 12) (dual of [531460, 531412, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(7) [i] based on
- linear OA(2745, 531441, F27, 12) (dual of [531441, 531396, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(2729, 531441, F27, 8) (dual of [531441, 531412, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(273, 19, F27, 3) (dual of [19, 16, 4]-code or 19-arc in PG(2,27) or 19-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
(36, 36+12, large)-Net in Base 27 — Upper bound on s
There is no (36, 48, large)-net in base 27, because
- 10 times m-reduction [i] would yield (36, 38, large)-net in base 27, but