Best Known (21, 21+7, s)-Nets in Base 81
(21, 21+7, 2796282)-Net over F81 — Constructive and digital
Digital (21, 28, 2796282)-net over F81, using
- net defined by OOA [i] based on linear OOA(8128, 2796282, F81, 9, 7) (dual of [(2796282, 9), 25166510, 8]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(8128, 2796283, F81, 3, 7) (dual of [(2796283, 3), 8388821, 8]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(813, 82, F81, 3, 3) (dual of [(82, 3), 243, 4]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;243,81) [i]
- linear OOA(8125, 2796201, F81, 3, 7) (dual of [(2796201, 3), 8388578, 8]-NRT-code), using
- OOA 3-folding [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523361 | 818−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(8125, large, F81, 7) (dual of [large, large−25, 8]-code), using
- linear OOA(813, 82, F81, 3, 3) (dual of [(82, 3), 243, 4]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(8128, 2796283, F81, 3, 7) (dual of [(2796283, 3), 8388821, 8]-NRT-code), using
(21, 21+7, large)-Net over F81 — Digital
Digital (21, 28, large)-net over F81, using
- 1 times m-reduction [i] based on digital (21, 29, large)-net over F81, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 21523360 | 814−1, defining interval I = [0,7], and designed minimum distance d ≥ |I|+1 = 9 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(8129, large, F81, 8) (dual of [large, large−29, 9]-code), using
(21, 21+7, large)-Net in Base 81 — Upper bound on s
There is no (21, 28, large)-net in base 81, because
- 5 times m-reduction [i] would yield (21, 23, large)-net in base 81, but