Best Known (54, 54+26, s)-Nets in Base 27
(54, 54+26, 1515)-Net over F27 — Constructive and digital
Digital (54, 80, 1515)-net over F27, using
- 271 times duplication [i] based on digital (53, 79, 1515)-net over F27, using
- net defined by OOA [i] based on linear OOA(2779, 1515, F27, 26, 26) (dual of [(1515, 26), 39311, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(2779, 19695, F27, 26) (dual of [19695, 19616, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2779, 19698, F27, 26) (dual of [19698, 19619, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(21) [i] based on
- linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(2764, 19683, F27, 22) (dual of [19683, 19619, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [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(25) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(2779, 19698, F27, 26) (dual of [19698, 19619, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(2779, 19695, F27, 26) (dual of [19695, 19616, 27]-code), using
- net defined by OOA [i] based on linear OOA(2779, 1515, F27, 26, 26) (dual of [(1515, 26), 39311, 27]-NRT-code), using
(54, 54+26, 19395)-Net over F27 — Digital
Digital (54, 80, 19395)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2780, 19395, F27, 26) (dual of [19395, 19315, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(2780, 19702, F27, 26) (dual of [19702, 19622, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- linear OA(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [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(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(25) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(2780, 19702, F27, 26) (dual of [19702, 19622, 27]-code), using
(54, 54+26, large)-Net in Base 27 — Upper bound on s
There is no (54, 80, large)-net in base 27, because
- 24 times m-reduction [i] would yield (54, 56, large)-net in base 27, but