Best Known (81, 81+21, s)-Nets in Base 27
(81, 81+21, 838860)-Net over F27 — Constructive and digital
Digital (81, 102, 838860)-net over F27, using
- 271 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
(81, 81+21, large)-Net over F27 — Digital
Digital (81, 102, large)-net over F27, using
- 271 times duplication [i] based on digital (80, 101, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
(81, 81+21, large)-Net in Base 27 — Upper bound on s
There is no (81, 102, large)-net in base 27, because
- 19 times m-reduction [i] would yield (81, 83, large)-net in base 27, but