Best Known (144, 161, s)-Nets in Base 2
(144, 161, 131072)-Net over F2 — Constructive and digital
Digital (144, 161, 131072)-net over F2, using
- net defined by OOA [i] based on linear OOA(2161, 131072, F2, 17, 17) (dual of [(131072, 17), 2228063, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2161, 1048577, F2, 17) (dual of [1048577, 1048416, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1048577 | 240−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- OOA 8-folding and stacking with additional row [i] based on linear OA(2161, 1048577, F2, 17) (dual of [1048577, 1048416, 18]-code), using
(144, 161, 174762)-Net over F2 — Digital
Digital (144, 161, 174762)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2161, 174762, F2, 6, 17) (dual of [(174762, 6), 1048411, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2161, 1048572, F2, 17) (dual of [1048572, 1048411, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 220−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2161, 1048576, F2, 17) (dual of [1048576, 1048415, 18]-code), using
- OOA 6-folding [i] based on linear OA(2161, 1048572, F2, 17) (dual of [1048572, 1048411, 18]-code), using
(144, 161, 3947196)-Net in Base 2 — Upper bound on s
There is no (144, 161, 3947197)-net in base 2, because
- 1 times m-reduction [i] would yield (144, 160, 3947197)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 461504 024073 741482 467447 509122 735173 916055 450205 > 2160 [i]