Best Known (84−13, 84, s)-Nets in Base 27
(84−13, 84, 1575251)-Net over F27 — Constructive and digital
Digital (71, 84, 1575251)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (17, 23, 177151)-net over F27, using
- net defined by OOA [i] based on linear OOA(2723, 177151, F27, 6, 6) (dual of [(177151, 6), 1062883, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2723, 531453, F27, 6) (dual of [531453, 531430, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(2723, 531455, F27, 6) (dual of [531455, 531432, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(2) [i] based on
- 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(279, 531441, F27, 3) (dual of [531441, 531432, 4]-code or 531441-cap in PG(8,27)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(272, 14, F27, 2) (dual of [14, 12, 3]-code or 14-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(5) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(2723, 531455, F27, 6) (dual of [531455, 531432, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(2723, 531453, F27, 6) (dual of [531453, 531430, 7]-code), using
- net defined by OOA [i] based on linear OOA(2723, 177151, F27, 6, 6) (dual of [(177151, 6), 1062883, 7]-NRT-code), using
- digital (48, 61, 1398100)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (17, 23, 177151)-net over F27, using
(84−13, 84, large)-Net over F27 — Digital
Digital (71, 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
(84−13, 84, large)-Net in Base 27 — Upper bound on s
There is no (71, 84, large)-net in base 27, because
- 11 times m-reduction [i] would yield (71, 73, large)-net in base 27, but