Best Known (72, 72+15, s)-Nets in Base 27
(72, 72+15, 1199127)-Net over F27 — Constructive and digital
Digital (72, 87, 1199127)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (9, 16, 756)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 28)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 0, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 1, 28)-net over F27 (see above)
- digital (0, 2, 28)-net over F27, using
- digital (0, 3, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F27 with g(F) = 0 and N(F) ≥ 28, using
- the rational function field F27(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 27)-sequence over F27, using
- digital (0, 7, 28)-net over F27, using
- net from sequence [i] based on digital (0, 27)-sequence over F27 (see above)
- digital (0, 0, 28)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- 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 (9, 16, 756)-net over F27, using
(72, 72+15, large)-Net over F27 — Digital
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
(72, 72+15, large)-Net in Base 27 — Upper bound on s
There is no (72, 87, large)-net in base 27, because
- 13 times m-reduction [i] would yield (72, 74, large)-net in base 27, but