Best Known (224, 224+28, s)-Nets in Base 2
(224, 224+28, 18724)-Net over F2 — Constructive and digital
Digital (224, 252, 18724)-net over F2, using
- net defined by OOA [i] based on linear OOA(2252, 18724, F2, 28, 28) (dual of [(18724, 28), 524020, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(2252, 262136, F2, 28) (dual of [262136, 261884, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2252, 262144, F2, 28) (dual of [262144, 261892, 29]-code), using
- 1 times truncation [i] based on linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2252, 262144, F2, 28) (dual of [262144, 261892, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(2252, 262136, F2, 28) (dual of [262136, 261884, 29]-code), using
(224, 224+28, 37449)-Net over F2 — Digital
Digital (224, 252, 37449)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2252, 37449, F2, 7, 28) (dual of [(37449, 7), 261891, 29]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2252, 262143, F2, 28) (dual of [262143, 261891, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2252, 262144, F2, 28) (dual of [262144, 261892, 29]-code), using
- 1 times truncation [i] based on linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- 1 times truncation [i] based on linear OA(2253, 262145, F2, 29) (dual of [262145, 261892, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2252, 262144, F2, 28) (dual of [262144, 261892, 29]-code), using
- OOA 7-folding [i] based on linear OA(2252, 262143, F2, 28) (dual of [262143, 261891, 29]-code), using
(224, 224+28, 1584864)-Net in Base 2 — Upper bound on s
There is no (224, 252, 1584865)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 7237 059393 396539 037035 363250 105589 788600 625240 263641 288293 596983 535748 276688 > 2252 [i]