Best Known (44−6, 44, s)-Nets in Base 3
(44−6, 44, 59049)-Net over F3 — Constructive and digital
Digital (38, 44, 59049)-net over F3, using
- net defined by OOA [i] based on linear OOA(344, 59049, F3, 6, 6) (dual of [(59049, 6), 354250, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(344, 59049, F3, 5, 6) (dual of [(59049, 5), 295201, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(344, 177147, F3, 6) (dual of [177147, 177103, 7]-code), using
- 1 times truncation [i] based on linear OA(345, 177148, F3, 7) (dual of [177148, 177103, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(345, 177148, F3, 7) (dual of [177148, 177103, 8]-code), using
- OA 3-folding and stacking [i] based on linear OA(344, 177147, F3, 6) (dual of [177147, 177103, 7]-code), using
- appending kth column [i] based on linear OOA(344, 59049, F3, 5, 6) (dual of [(59049, 5), 295201, 7]-NRT-code), using
(44−6, 44, 148959)-Net over F3 — Digital
Digital (38, 44, 148959)-net over F3, using
- net defined by OOA [i] based on linear OOA(344, 148959, F3, 6, 6) (dual of [(148959, 6), 893710, 7]-NRT-code), using
- appending kth column [i] based on linear OOA(344, 148959, F3, 5, 6) (dual of [(148959, 5), 744751, 7]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 148959, F3, 6) (dual of [148959, 148915, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 177147, F3, 6) (dual of [177147, 177103, 7]-code), using
- 1 times truncation [i] based on linear OA(345, 177148, F3, 7) (dual of [177148, 177103, 8]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,3], and minimum distance d ≥ |{−3,−2,…,3}|+1 = 8 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(345, 177148, F3, 7) (dual of [177148, 177103, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(344, 177147, F3, 6) (dual of [177147, 177103, 7]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(344, 148959, F3, 6) (dual of [148959, 148915, 7]-code), using
- appending kth column [i] based on linear OOA(344, 148959, F3, 5, 6) (dual of [(148959, 5), 744751, 7]-NRT-code), using
(44−6, 44, large)-Net in Base 3 — Upper bound on s
There is no (38, 44, large)-net in base 3, because
- 4 times m-reduction [i] would yield (38, 40, large)-net in base 3, but