Best Known (56−17, 56, s)-Nets in Base 27
(56−17, 56, 2463)-Net over F27 — Constructive and digital
Digital (39, 56, 2463)-net over F27, using
- 272 times duplication [i] based on digital (37, 54, 2463)-net over F27, using
- net defined by OOA [i] based on linear OOA(2754, 2463, F27, 17, 17) (dual of [(2463, 17), 41817, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2754, 19705, F27, 17) (dual of [19705, 19651, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2754, 19707, F27, 17) (dual of [19707, 19653, 18]-code), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- linear OA(2749, 19684, F27, 17) (dual of [19684, 19635, 18]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- linear OA(2731, 19684, F27, 11) (dual of [19684, 19653, 12]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- linear OA(275, 23, F27, 5) (dual of [23, 18, 6]-code or 23-arc in PG(4,27)), using
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- Reed–Solomon code RS(22,27) [i]
- discarding factors / shortening the dual code based on linear OA(275, 27, F27, 5) (dual of [27, 22, 6]-code or 27-arc in PG(4,27)), using
- construction X applied to C([0,8]) ⊂ C([0,5]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2754, 19707, F27, 17) (dual of [19707, 19653, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2754, 19705, F27, 17) (dual of [19705, 19651, 18]-code), using
- net defined by OOA [i] based on linear OOA(2754, 2463, F27, 17, 17) (dual of [(2463, 17), 41817, 18]-NRT-code), using
(56−17, 56, 26759)-Net over F27 — Digital
Digital (39, 56, 26759)-net over F27, using
(56−17, 56, large)-Net in Base 27 — Upper bound on s
There is no (39, 56, large)-net in base 27, because
- 15 times m-reduction [i] would yield (39, 41, large)-net in base 27, but