Best Known (181−21, 181, s)-Nets in Base 2
(181−21, 181, 26214)-Net over F2 — Constructive and digital
Digital (160, 181, 26214)-net over F2, using
- net defined by OOA [i] based on linear OOA(2181, 26214, F2, 21, 21) (dual of [(26214, 21), 550313, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2181, 262141, F2, 21) (dual of [262141, 261960, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2181, 262141, F2, 21) (dual of [262141, 261960, 22]-code), using
(181−21, 181, 37449)-Net over F2 — Digital
Digital (160, 181, 37449)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2181, 37449, F2, 7, 21) (dual of [(37449, 7), 261962, 22]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2181, 262143, F2, 21) (dual of [262143, 261962, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using
- OOA 7-folding [i] based on linear OA(2181, 262143, F2, 21) (dual of [262143, 261962, 22]-code), using
(181−21, 181, 1187164)-Net in Base 2 — Upper bound on s
There is no (160, 181, 1187165)-net in base 2, because
- 1 times m-reduction [i] would yield (160, 180, 1187165)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 532501 307237 576087 704892 289518 843074 042556 750720 835578 > 2180 [i]