Best Known (201−21, 201, s)-Nets in Base 2
(201−21, 201, 104857)-Net over F2 — Constructive and digital
Digital (180, 201, 104857)-net over F2, using
- net defined by OOA [i] based on linear OOA(2201, 104857, F2, 21, 21) (dual of [(104857, 21), 2201796, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2201, 1048571, F2, 21) (dual of [1048571, 1048370, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2201, 1048571, F2, 21) (dual of [1048571, 1048370, 22]-code), using
(201−21, 201, 149796)-Net over F2 — Digital
Digital (180, 201, 149796)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2201, 149796, F2, 7, 21) (dual of [(149796, 7), 1048371, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2201, 1048572, F2, 21) (dual of [1048572, 1048371, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2201, 1048576, F2, 21) (dual of [1048576, 1048375, 22]-code), using
- OOA 7-folding [i] based on linear OA(2201, 1048572, F2, 21) (dual of [1048572, 1048371, 22]-code), using
(201−21, 201, 4748701)-Net in Base 2 — Upper bound on s
There is no (180, 201, 4748702)-net in base 2, because
- 1 times m-reduction [i] would yield (180, 200, 4748702)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 606939 015821 007457 331120 625317 637385 527703 835820 210923 724434 > 2200 [i]