Best Known (109−21, 109, s)-Nets in Base 27
(109−21, 109, 838860)-Net over F27 — Constructive and digital
Digital (88, 109, 838860)-net over F27, using
- 278 times duplication [i] based on digital (80, 101, 838860)-net over F27, using
- net defined by OOA [i] based on linear OOA(27101, 838860, F27, 21, 21) (dual of [(838860, 21), 17615959, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(27101, 8388601, F27, 21) (dual of [8388601, 8388500, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(27101, 8388601, F27, 21) (dual of [8388601, 8388500, 22]-code), using
- net defined by OOA [i] based on linear OOA(27101, 838860, F27, 21, 21) (dual of [(838860, 21), 17615959, 22]-NRT-code), using
(109−21, 109, large)-Net over F27 — Digital
Digital (88, 109, large)-net over F27, using
- 273 times duplication [i] based on digital (85, 106, large)-net over F27, using
- t-expansion [i] based on digital (84, 106, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- t-expansion [i] based on digital (84, 106, large)-net over F27, using
(109−21, 109, large)-Net in Base 27 — Upper bound on s
There is no (88, 109, large)-net in base 27, because
- 19 times m-reduction [i] would yield (88, 90, large)-net in base 27, but