Best Known (84, 101, s)-Nets in Base 27
(84, 101, 1049331)-Net over F27 — Constructive and digital
Digital (84, 101, 1049331)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (12, 20, 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, 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, 2, 28)-net over F27 (see above)
- digital (0, 4, 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, 8, 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 (64, 81, 1048575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- digital (12, 20, 756)-net over F27, using
(84, 101, 1050215)-Net in Base 27 — Constructive
(84, 101, 1050215)-net in base 27, using
- (u, u+v)-construction [i] based on
- (12, 20, 1640)-net in base 27, using
- base change [i] based on digital (7, 15, 1640)-net over F81, using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 6560 = 812−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- discarding factors / shortening the dual code based on linear OA(8115, 6561, F81, 8) (dual of [6561, 6546, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(8115, 6560, F81, 8) (dual of [6560, 6545, 9]-code), using
- net defined by OOA [i] based on linear OOA(8115, 1640, F81, 8, 8) (dual of [(1640, 8), 13105, 9]-NRT-code), using
- base change [i] based on digital (7, 15, 1640)-net over F81, using
- digital (64, 81, 1048575)-net over F27, using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2781, large, F27, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2781, 8388601, F27, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(2781, 1048575, F27, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- (12, 20, 1640)-net in base 27, using
(84, 101, large)-Net over F27 — Digital
Digital (84, 101, large)-net over F27, using
- 5 times m-reduction [i] based on digital (84, 106, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(27106, large, F27, 22) (dual of [large, large−106, 23]-code), using
(84, 101, large)-Net in Base 27 — Upper bound on s
There is no (84, 101, large)-net in base 27, because
- 15 times m-reduction [i] would yield (84, 86, large)-net in base 27, but