Best Known (41, 41+19, s)-Nets in Base 27
(41, 41+19, 2189)-Net over F27 — Constructive and digital
Digital (41, 60, 2189)-net over F27, using
- 271 times duplication [i] based on digital (40, 59, 2189)-net over F27, using
- net defined by OOA [i] based on linear OOA(2759, 2189, F27, 19, 19) (dual of [(2189, 19), 41532, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2759, 19702, F27, 19) (dual of [19702, 19643, 20]-code), using
- construction X applied to Ce(18) ⊂ Ce(13) [i] based on
- linear OA(2755, 19683, F27, 19) (dual of [19683, 19628, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2740, 19683, F27, 14) (dual of [19683, 19643, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [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(18) ⊂ Ce(13) [i] based on
- OOA 9-folding and stacking with additional row [i] based on linear OA(2759, 19702, F27, 19) (dual of [19702, 19643, 20]-code), using
- net defined by OOA [i] based on linear OOA(2759, 2189, F27, 19, 19) (dual of [(2189, 19), 41532, 20]-NRT-code), using
(41, 41+19, 19707)-Net over F27 — Digital
Digital (41, 60, 19707)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2760, 19707, F27, 19) (dual of [19707, 19647, 20]-code), using
- construction X applied to C([0,9]) ⊂ C([0,6]) [i] based on
- linear OA(2755, 19684, F27, 19) (dual of [19684, 19629, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2737, 19684, F27, 13) (dual of [19684, 19647, 14]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (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([0,9]) ⊂ C([0,6]) [i] based on
(41, 41+19, large)-Net in Base 27 — Upper bound on s
There is no (41, 60, large)-net in base 27, because
- 17 times m-reduction [i] would yield (41, 43, large)-net in base 27, but