Best Known (69, 104, s)-Nets in Base 27
(69, 104, 1158)-Net over F27 — Constructive and digital
Digital (69, 104, 1158)-net over F27, using
- 273 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
(69, 104, 14842)-Net over F27 — Digital
Digital (69, 104, 14842)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27104, 14842, F27, 35) (dual of [14842, 14738, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(27104, 19691, F27, 35) (dual of [19691, 19587, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,16]) [i] based on
- linear OA(27103, 19684, F27, 35) (dual of [19684, 19581, 36]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,17], and minimum distance d ≥ |{−17,−16,…,17}|+1 = 36 (BCH-bound) [i]
- linear OA(2797, 19684, F27, 33) (dual of [19684, 19587, 34]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [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 C([0,17]) ⊂ C([0,16]) [i] based on
- discarding factors / shortening the dual code based on linear OA(27104, 19691, F27, 35) (dual of [19691, 19587, 36]-code), using
(69, 104, large)-Net in Base 27 — Upper bound on s
There is no (69, 104, large)-net in base 27, because
- 33 times m-reduction [i] would yield (69, 71, large)-net in base 27, but