Best Known (86, 86+15, s)-Nets in Base 27
(86, 86+15, 1377227)-Net over F27 — Constructive and digital
Digital (86, 101, 1377227)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (23, 30, 178856)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (2, 5, 1708)-net over F27, using
- net defined by OOA [i] based on linear OOA(275, 1708, F27, 3, 3) (dual of [(1708, 3), 5119, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(275, 1708, F27, 2, 3) (dual of [(1708, 2), 3411, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(275, 1708, F27, 3, 3) (dual of [(1708, 3), 5119, 4]-NRT-code), using
- digital (18, 25, 177148)-net over F27, using
- net defined by OOA [i] based on linear OOA(2725, 177148, F27, 7, 7) (dual of [(177148, 7), 1240011, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2725, 531445, F27, 7) (dual of [531445, 531420, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(2725, 531441, F27, 7) (dual of [531441, 531416, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2721, 531441, F27, 6) (dual of [531441, 531420, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(270, 4, F27, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(270, s, F27, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- OOA 3-folding and stacking with additional row [i] based on linear OA(2725, 531445, F27, 7) (dual of [531445, 531420, 8]-code), using
- net defined by OOA [i] based on linear OOA(2725, 177148, F27, 7, 7) (dual of [(177148, 7), 1240011, 8]-NRT-code), using
- digital (2, 5, 1708)-net over F27, using
- (u, u+v)-construction [i] based on
- 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 (23, 30, 178856)-net over F27, using
(86, 86+15, large)-Net over F27 — Digital
Digital (86, 101, large)-net over F27, using
- t-expansion [i] based on digital (84, 101, large)-net over F27, using
- 5 times m-reduction [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
- 5 times m-reduction [i] based on digital (84, 106, large)-net over F27, using
(86, 86+15, large)-Net in Base 27 — Upper bound on s
There is no (86, 101, large)-net in base 27, because
- 13 times m-reduction [i] would yield (86, 88, large)-net in base 27, but