Best Known (84−15, 84, s)-Nets in Base 27
(84−15, 84, 1198614)-Net over F27 — Constructive and digital
Digital (69, 84, 1198614)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (6, 13, 243)-net over F27, using
- net defined by OOA [i] based on linear OOA(2713, 243, F27, 7, 7) (dual of [(243, 7), 1688, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding and stacking with additional row [i] based on linear OA(2713, 730, F27, 7) (dual of [730, 717, 8]-code), using
- net defined by OOA [i] based on linear OOA(2713, 243, F27, 7, 7) (dual of [(243, 7), 1688, 8]-NRT-code), using
- digital (56, 71, 1198371)-net over F27, using
- net defined by OOA [i] based on linear OOA(2771, 1198371, F27, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2771, 8388598, F27, 15) (dual of [8388598, 8388527, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2771, large, F27, 15) (dual of [large, large−71, 16]-code), using
- OOA 7-folding and stacking with additional row [i] based on linear OA(2771, 8388598, F27, 15) (dual of [8388598, 8388527, 16]-code), using
- net defined by OOA [i] based on linear OOA(2771, 1198371, F27, 15, 15) (dual of [(1198371, 15), 17975494, 16]-NRT-code), using
- digital (6, 13, 243)-net over F27, using
(84−15, 84, large)-Net over F27 — Digital
Digital (69, 84, large)-net over F27, using
- t-expansion [i] based on digital (68, 84, large)-net over F27, using
- 2 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- 2 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
(84−15, 84, large)-Net in Base 27 — Upper bound on s
There is no (69, 84, large)-net in base 27, because
- 13 times m-reduction [i] would yield (69, 71, large)-net in base 27, but