Best Known (44−8, 44, s)-Nets in Base 2
(44−8, 44, 512)-Net over F2 — Constructive and digital
Digital (36, 44, 512)-net over F2, using
- net defined by OOA [i] based on linear OOA(244, 512, F2, 8, 8) (dual of [(512, 8), 4052, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- 1 times truncation [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- OA 4-folding and stacking [i] based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
(44−8, 44, 874)-Net over F2 — Digital
Digital (36, 44, 874)-net over F2, using
- net defined by OOA [i] based on linear OOA(244, 874, F2, 8, 8) (dual of [(874, 8), 6948, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(244, 874, F2, 7, 8) (dual of [(874, 7), 6074, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 874, F2, 2, 8) (dual of [(874, 2), 1704, 9]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 1024, F2, 2, 8) (dual of [(1024, 2), 2004, 9]-NRT-code), using
- OOA 2-folding [i] based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- 1 times truncation [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 2049 | 222−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(245, 2049, F2, 9) (dual of [2049, 2004, 10]-code), using
- OOA 2-folding [i] based on linear OA(244, 2048, F2, 8) (dual of [2048, 2004, 9]-code), using
- discarding factors / shortening the dual code based on linear OOA(244, 1024, F2, 2, 8) (dual of [(1024, 2), 2004, 9]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(244, 874, F2, 2, 8) (dual of [(874, 2), 1704, 9]-NRT-code), using
- appending kth column [i] based on linear OOA(244, 874, F2, 7, 8) (dual of [(874, 7), 6074, 9]-NRT-code), using
(44−8, 44, 4527)-Net in Base 2 — Upper bound on s
There is no (36, 44, 4528)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 17 600385 161629 > 244 [i]