Best Known (49, 49+24, s)-Nets in Base 27
(49, 49+24, 1641)-Net over F27 — Constructive and digital
Digital (49, 73, 1641)-net over F27, using
- 271 times duplication [i] based on digital (48, 72, 1641)-net over F27, using
- net defined by OOA [i] based on linear OOA(2772, 1641, F27, 24, 24) (dual of [(1641, 24), 39312, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(2772, 19692, F27, 24) (dual of [19692, 19620, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2772, 19694, F27, 24) (dual of [19694, 19622, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(20) [i] based on
- linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2761, 19683, F27, 21) (dual of [19683, 19622, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(23) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2772, 19694, F27, 24) (dual of [19694, 19622, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(2772, 19692, F27, 24) (dual of [19692, 19620, 25]-code), using
- net defined by OOA [i] based on linear OOA(2772, 1641, F27, 24, 24) (dual of [(1641, 24), 39312, 25]-NRT-code), using
(49, 49+24, 16829)-Net over F27 — Digital
Digital (49, 73, 16829)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2773, 16829, F27, 24) (dual of [16829, 16756, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(2773, 19698, F27, 24) (dual of [19698, 19625, 25]-code), using
- construction X applied to Ce(23) ⊂ Ce(19) [i] based on
- linear OA(2770, 19683, F27, 24) (dual of [19683, 19613, 25]-code), using an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- linear OA(2758, 19683, F27, 20) (dual of [19683, 19625, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [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(23) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(2773, 19698, F27, 24) (dual of [19698, 19625, 25]-code), using
(49, 49+24, large)-Net in Base 27 — Upper bound on s
There is no (49, 73, large)-net in base 27, because
- 22 times m-reduction [i] would yield (49, 51, large)-net in base 27, but