Best Known (87−14, 87, s)-Nets in Base 27
(87−14, 87, 1204935)-Net over F27 — Constructive and digital
Digital (73, 87, 1204935)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (14, 21, 6564)-net over F27, using
- net defined by OOA [i] based on linear OOA(2721, 6564, F27, 7, 7) (dual of [(6564, 7), 45927, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2721, 19693, F27, 7) (dual of [19693, 19672, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2721, 19694, F27, 7) (dual of [19694, 19673, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2719, 19683, F27, 7) (dual of [19683, 19664, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(2710, 19683, F27, 4) (dual of [19683, 19673, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 19682 = 273−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(272, 11, F27, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,27)), using
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- Reed–Solomon code RS(25,27) [i]
- discarding factors / shortening the dual code based on linear OA(272, 27, F27, 2) (dual of [27, 25, 3]-code or 27-arc in PG(1,27)), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(2721, 19694, F27, 7) (dual of [19694, 19673, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2721, 19693, F27, 7) (dual of [19693, 19672, 8]-code), using
- net defined by OOA [i] based on linear OOA(2721, 6564, F27, 7, 7) (dual of [(6564, 7), 45927, 8]-NRT-code), using
- digital (52, 66, 1198371)-net over F27, using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- discarding factors / shortening the dual code based on linear OA(2766, large, F27, 14) (dual of [large, large−66, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(2766, 8388597, F27, 14) (dual of [8388597, 8388531, 15]-code), using
- net defined by OOA [i] based on linear OOA(2766, 1198371, F27, 14, 14) (dual of [(1198371, 14), 16777128, 15]-NRT-code), using
- digital (14, 21, 6564)-net over F27, using
(87−14, 87, large)-Net over F27 — Digital
Digital (73, 87, large)-net over F27, using
- t-expansion [i] based on digital (72, 87, large)-net over F27, using
- 4 times m-reduction [i] based on digital (72, 91, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- 4 times m-reduction [i] based on digital (72, 91, large)-net over F27, using
(87−14, 87, large)-Net in Base 27 — Upper bound on s
There is no (73, 87, large)-net in base 27, because
- 12 times m-reduction [i] would yield (73, 75, large)-net in base 27, but