Best Known (106−35, 106, s)-Nets in Base 27
(106−35, 106, 1159)-Net over F27 — Constructive and digital
Digital (71, 106, 1159)-net over F27, using
- 271 times duplication [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
(106−35, 106, 18127)-Net over F27 — Digital
Digital (71, 106, 18127)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27106, 18127, F27, 35) (dual of [18127, 18021, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(27106, 19699, F27, 35) (dual of [19699, 19593, 36]-code), using
- construction X applied to C([0,17]) ⊂ C([0,15]) [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(2791, 19684, F27, 31) (dual of [19684, 19593, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 276−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [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 C([0,17]) ⊂ C([0,15]) [i] based on
- discarding factors / shortening the dual code based on linear OA(27106, 19699, F27, 35) (dual of [19699, 19593, 36]-code), using
(106−35, 106, large)-Net in Base 27 — Upper bound on s
There is no (71, 106, large)-net in base 27, because
- 33 times m-reduction [i] would yield (71, 73, large)-net in base 27, but