Best Known (67, 67+35, s)-Nets in Base 27
(67, 67+35, 1158)-Net over F27 — Constructive and digital
Digital (67, 102, 1158)-net over F27, using
- 271 times duplication [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
(67, 67+35, 12151)-Net over F27 — Digital
Digital (67, 102, 12151)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27102, 12151, F27, 35) (dual of [12151, 12049, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(27102, 19694, F27, 35) (dual of [19694, 19592, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(31) [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(2791, 19683, F27, 32) (dual of [19683, 19592, 33]-code), using an extension Ce(31) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,31], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(34) ⊂ Ce(31) [i] based on
- discarding factors / shortening the dual code based on linear OA(27102, 19694, F27, 35) (dual of [19694, 19592, 36]-code), using
(67, 67+35, large)-Net in Base 27 — Upper bound on s
There is no (67, 102, large)-net in base 27, because
- 33 times m-reduction [i] would yield (67, 69, large)-net in base 27, but