Best Known (70, 85, s)-Nets in Base 27
(70, 85, 1198615)-Net over F27 — Constructive and digital
Digital (70, 85, 1198615)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (7, 14, 244)-net over F27, using
- net defined by OOA [i] based on linear OOA(2714, 244, F27, 7, 7) (dual of [(244, 7), 1694, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2714, 733, F27, 7) (dual of [733, 719, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2714, 735, F27, 7) (dual of [735, 721, 8]-code), using
- construction X applied to C([0,3]) ⊂ C([0,2]) [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]
- linear OA(279, 730, F27, 5) (dual of [730, 721, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 730 | 274−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(271, 5, F27, 1) (dual of [5, 4, 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 C([0,3]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2714, 735, F27, 7) (dual of [735, 721, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2714, 733, F27, 7) (dual of [733, 719, 8]-code), using
- net defined by OOA [i] based on linear OOA(2714, 244, F27, 7, 7) (dual of [(244, 7), 1694, 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 (7, 14, 244)-net over F27, using
(70, 85, large)-Net over F27 — Digital
Digital (70, 85, large)-net over F27, using
- t-expansion [i] based on digital (68, 85, large)-net over F27, using
- 1 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
- 1 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
(70, 85, large)-Net in Base 27 — Upper bound on s
There is no (70, 85, large)-net in base 27, because
- 13 times m-reduction [i] would yield (70, 72, large)-net in base 27, but