Best Known (71, 71+15, s)-Nets in Base 27
(71, 71+15, 1198616)-Net over F27 — Constructive and digital
Digital (71, 86, 1198616)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (8, 15, 245)-net over F27, using
- net defined by OOA [i] based on linear OOA(2715, 245, F27, 7, 7) (dual of [(245, 7), 1700, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2715, 736, F27, 7) (dual of [736, 721, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(2715, 737, F27, 7) (dual of [737, 722, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(3) [i] based on
- linear OA(2713, 729, F27, 7) (dual of [729, 716, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(277, 729, F27, 4) (dual of [729, 722, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 728 = 272−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(272, 8, F27, 2) (dual of [8, 6, 3]-code or 8-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(2715, 737, F27, 7) (dual of [737, 722, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(2715, 736, F27, 7) (dual of [736, 721, 8]-code), using
- net defined by OOA [i] based on linear OOA(2715, 245, F27, 7, 7) (dual of [(245, 7), 1700, 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 (8, 15, 245)-net over F27, using
(71, 71+15, large)-Net over F27 — Digital
Digital (71, 86, large)-net over F27, using
- t-expansion [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
(71, 71+15, large)-Net in Base 27 — Upper bound on s
There is no (71, 86, large)-net in base 27, because
- 13 times m-reduction [i] would yield (71, 73, large)-net in base 27, but