Best Known (82, 82+19, s)-Nets in Base 27
(82, 82+19, 932104)-Net over F27 — Constructive and digital
Digital (82, 101, 932104)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 38)-net over F27, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 1 and N(F) ≥ 38, using
- net from sequence [i] based on digital (1, 37)-sequence over F27, using
- digital (72, 91, 932066)-net over F27, using
- net defined by OOA [i] based on linear OOA(2791, 932066, F27, 19, 19) (dual of [(932066, 19), 17709163, 20]-NRT-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2791, 8388595, F27, 19) (dual of [8388595, 8388504, 20]-code), using
- discarding factors / shortening the dual code 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]
- discarding factors / shortening the dual code based on linear OA(2791, large, F27, 19) (dual of [large, large−91, 20]-code), using
- OOA 9-folding and stacking with additional row [i] based on linear OA(2791, 8388595, F27, 19) (dual of [8388595, 8388504, 20]-code), using
- net defined by OOA [i] based on linear OOA(2791, 932066, F27, 19, 19) (dual of [(932066, 19), 17709163, 20]-NRT-code), using
- digital (1, 10, 38)-net over F27, using
(82, 82+19, large)-Net over F27 — Digital
Digital (82, 101, large)-net over F27, using
- t-expansion [i] based on digital (80, 101, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27101, large, F27, 21) (dual of [large, large−101, 22]-code), using
(82, 82+19, large)-Net in Base 27 — Upper bound on s
There is no (82, 101, large)-net in base 27, because
- 17 times m-reduction [i] would yield (82, 84, large)-net in base 27, but