Best Known (51, 51+10, s)-Nets in Base 27
(51, 51+10, 1687953)-Net over F27 — Constructive and digital
Digital (51, 61, 1687953)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (10, 15, 10233)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 379)-net over F27, using
- s-reduction based on digital (0, 0, s)-net over F27 with arbitrarily large s, using
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 0, 379)-net over F27 (see above)
- digital (0, 1, 379)-net over F27, using
- s-reduction based on digital (0, 1, s)-net over F27 with arbitrarily large s, using
- digital (0, 1, 379)-net over F27 (see above)
- digital (0, 1, 379)-net over F27 (see above)
- digital (1, 3, 379)-net over F27, using
- s-reduction based on digital (1, 3, 757)-net over F27, using
- digital (4, 9, 379)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (0, 2, 28)-net over F27, using
- digital (2, 7, 351)-net over F27, using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(277, 703, F27, 5) (dual of [703, 696, 6]-code), using
- net defined by OOA [i] based on linear OOA(277, 351, F27, 5, 5) (dual of [(351, 5), 1748, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- digital (0, 0, 379)-net over F27, using
- generalized (u, u+v)-construction [i] based on
- digital (36, 46, 1677720)-net over F27, using
- net defined by OOA [i] based on linear OOA(2746, 1677720, F27, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(2746, 8388600, F27, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(2746, large, F27, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(2746, 8388600, F27, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(2746, 1677720, F27, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (10, 15, 10233)-net over F27, using
(51, 51+10, large)-Net over F27 — Digital
Digital (51, 61, large)-net over F27, using
- t-expansion [i] based on digital (48, 61, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] 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]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
(51, 51+10, large)-Net in Base 27 — Upper bound on s
There is no (51, 61, large)-net in base 27, because
- 8 times m-reduction [i] would yield (51, 53, large)-net in base 27, but