Best Known (66, 66+34, s)-Nets in Base 27
(66, 66+34, 1158)-Net over F27 — Constructive and digital
Digital (66, 100, 1158)-net over F27, using
- 1 times m-reduction [i] based on digital (66, 101, 1158)-net over F27, using
- net defined by OOA [i] based on linear OOA(27101, 1158, F27, 35, 35) (dual of [(1158, 35), 40429, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(27101, 19687, F27, 35) (dual of [19687, 19586, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(27101, 19690, F27, 35) (dual of [19690, 19589, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- linear OA(27100, 19683, F27, 35) (dual of [19683, 19583, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2794, 19683, F27, 33) (dual of [19683, 19589, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(271, 7, F27, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(34) ⊂ Ce(32) [i] based on
- discarding factors / shortening the dual code based on linear OA(27101, 19690, F27, 35) (dual of [19690, 19589, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(27101, 19687, F27, 35) (dual of [19687, 19586, 36]-code), using
- net defined by OOA [i] based on linear OOA(27101, 1158, F27, 35, 35) (dual of [(1158, 35), 40429, 36]-NRT-code), using
(66, 66+34, 13172)-Net over F27 — Digital
Digital (66, 100, 13172)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27100, 13172, F27, 34) (dual of [13172, 13072, 35]-code), using
- discarding factors / shortening the dual code based on linear OA(27100, 19698, F27, 34) (dual of [19698, 19598, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(29) [i] based on
- linear OA(2797, 19683, F27, 34) (dual of [19683, 19586, 35]-code), using an extension Ce(33) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,33], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2785, 19683, F27, 30) (dual of [19683, 19598, 31]-code), using an extension Ce(29) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,29], and designed minimum distance d ≥ |I|+1 = 30 [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(33) ⊂ Ce(29) [i] based on
- discarding factors / shortening the dual code based on linear OA(27100, 19698, F27, 34) (dual of [19698, 19598, 35]-code), using
(66, 66+34, large)-Net in Base 27 — Upper bound on s
There is no (66, 100, large)-net in base 27, because
- 32 times m-reduction [i] would yield (66, 68, large)-net in base 27, but