Best Known (53−9, 53, s)-Nets in Base 2
(53−9, 53, 2048)-Net over F2 — Constructive and digital
Digital (44, 53, 2048)-net over F2, using
- net defined by OOA [i] based on linear OOA(253, 2048, F2, 9, 9) (dual of [(2048, 9), 18379, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(253, 2048, F2, 8, 9) (dual of [(2048, 8), 16331, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(253, 8193, F2, 9) (dual of [8193, 8140, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 4-folding and stacking with additional row [i] based on linear OA(253, 8193, F2, 9) (dual of [8193, 8140, 10]-code), using
- appending kth column [i] based on linear OOA(253, 2048, F2, 8, 9) (dual of [(2048, 8), 16331, 10]-NRT-code), using
(53−9, 53, 2661)-Net over F2 — Digital
Digital (44, 53, 2661)-net over F2, using
- net defined by OOA [i] based on linear OOA(253, 2661, F2, 9, 9) (dual of [(2661, 9), 23896, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(253, 2661, F2, 8, 9) (dual of [(2661, 8), 21235, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(253, 2661, F2, 3, 9) (dual of [(2661, 3), 7930, 10]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(253, 2731, F2, 3, 9) (dual of [(2731, 3), 8140, 10]-NRT-code), using
- OOA 3-folding [i] based on linear OA(253, 8193, F2, 9) (dual of [8193, 8140, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- OOA 3-folding [i] based on linear OA(253, 8193, F2, 9) (dual of [8193, 8140, 10]-code), using
- discarding factors / shortening the dual code based on linear OOA(253, 2731, F2, 3, 9) (dual of [(2731, 3), 8140, 10]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(253, 2661, F2, 3, 9) (dual of [(2661, 3), 7930, 10]-NRT-code), using
- appending kth column [i] based on linear OOA(253, 2661, F2, 8, 9) (dual of [(2661, 8), 21235, 10]-NRT-code), using
(53−9, 53, 18126)-Net in Base 2 — Upper bound on s
There is no (44, 53, 18127)-net in base 2, because
- 1 times m-reduction [i] would yield (44, 52, 18127)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 4504 218355 950673 > 252 [i]