Best Known (70, 70+34, s)-Nets in Base 27
(70, 70+34, 1159)-Net over F27 — Constructive and digital
Digital (70, 104, 1159)-net over F27, using
- 1 times m-reduction [i] based on digital (70, 105, 1159)-net over F27, using
- net defined by OOA [i] based on linear OOA(27105, 1159, F27, 35, 35) (dual of [(1159, 35), 40460, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(27105, 19704, F27, 35) (dual of [19704, 19599, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(27105, 19706, F27, 35) (dual of [19706, 19601, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(28) [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(2782, 19683, F27, 29) (dual of [19683, 19601, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [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 Ce(34) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(27105, 19706, F27, 35) (dual of [19706, 19601, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(27105, 19704, F27, 35) (dual of [19704, 19599, 36]-code), using
- net defined by OOA [i] based on linear OOA(27105, 1159, F27, 35, 35) (dual of [(1159, 35), 40460, 36]-NRT-code), using
(70, 70+34, 19711)-Net over F27 — Digital
Digital (70, 104, 19711)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27104, 19711, F27, 34) (dual of [19711, 19607, 35]-code), using
- construction X applied to Ce(33) ⊂ Ce(25) [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(2776, 19683, F27, 26) (dual of [19683, 19607, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(277, 28, F27, 7) (dual of [28, 21, 8]-code or 28-arc in PG(6,27)), using
- extended Reed–Solomon code RSe(21,27) [i]
- the expurgated narrow-sense BCH-code C(I) with length 28 | 272−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- construction X applied to Ce(33) ⊂ Ce(25) [i] based on
(70, 70+34, large)-Net in Base 27 — Upper bound on s
There is no (70, 104, large)-net in base 27, because
- 32 times m-reduction [i] would yield (70, 72, large)-net in base 27, but