Best Known (70, 70+37, s)-Nets in Base 27
(70, 70+37, 1093)-Net over F27 — Constructive and digital
Digital (70, 107, 1093)-net over F27, using
- 271 times duplication [i] based on digital (69, 106, 1093)-net over F27, using
- net defined by OOA [i] based on linear OOA(27106, 1093, F27, 37, 37) (dual of [(1093, 37), 40335, 38]-NRT-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(27106, 19675, F27, 37) (dual of [19675, 19569, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(27106, 19683, F27, 37) (dual of [19683, 19577, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- discarding factors / shortening the dual code based on linear OA(27106, 19683, F27, 37) (dual of [19683, 19577, 38]-code), using
- OOA 18-folding and stacking with additional row [i] based on linear OA(27106, 19675, F27, 37) (dual of [19675, 19569, 38]-code), using
- net defined by OOA [i] based on linear OOA(27106, 1093, F27, 37, 37) (dual of [(1093, 37), 40335, 38]-NRT-code), using
(70, 70+37, 11551)-Net over F27 — Digital
Digital (70, 107, 11551)-net over F27, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(27107, 11551, F27, 37) (dual of [11551, 11444, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(27107, 19690, F27, 37) (dual of [19690, 19583, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(34) [i] based on
- linear OA(27106, 19683, F27, 37) (dual of [19683, 19577, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- 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(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(36) ⊂ Ce(34) [i] based on
- discarding factors / shortening the dual code based on linear OA(27107, 19690, F27, 37) (dual of [19690, 19583, 38]-code), using
(70, 70+37, large)-Net in Base 27 — Upper bound on s
There is no (70, 107, large)-net in base 27, because
- 35 times m-reduction [i] would yield (70, 72, large)-net in base 27, but