Best Known (45, 45+21, s)-Nets in Base 27
(45, 45+21, 1970)-Net over F27 — Constructive and digital
Digital (45, 66, 1970)-net over F27, using
- 271 times duplication [i] based on digital (44, 65, 1970)-net over F27, using
- net defined by OOA [i] based on linear OOA(2765, 1970, F27, 21, 21) (dual of [(1970, 21), 41305, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2765, 19701, F27, 21) (dual of [19701, 19636, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2765, 19702, F27, 21) (dual of [19702, 19637, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(15) [i] based on
- 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(2746, 19683, F27, 16) (dual of [19683, 19637, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(2765, 19702, F27, 21) (dual of [19702, 19637, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2765, 19701, F27, 21) (dual of [19701, 19636, 22]-code), using
- net defined by OOA [i] based on linear OOA(2765, 1970, F27, 21, 21) (dual of [(1970, 21), 41305, 22]-NRT-code), using
(45, 45+21, 19707)-Net over F27 — Digital
Digital (45, 66, 19707)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2766, 19707, F27, 21) (dual of [19707, 19641, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,7]) [i] based on
- linear OA(2761, 19684, F27, 21) (dual of [19684, 19623, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- linear OA(2743, 19684, F27, 15) (dual of [19684, 19641, 16]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (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([0,10]) ⊂ C([0,7]) [i] based on
(45, 45+21, large)-Net in Base 27 — Upper bound on s
There is no (45, 66, large)-net in base 27, because
- 19 times m-reduction [i] would yield (45, 47, large)-net in base 27, but