Best Known (157−25, 157, s)-Nets in Base 2
(157−25, 157, 682)-Net over F2 — Constructive and digital
Digital (132, 157, 682)-net over F2, using
- net defined by OOA [i] based on linear OOA(2157, 682, F2, 25, 25) (dual of [(682, 25), 16893, 26]-NRT-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2157, 8185, F2, 25) (dual of [8185, 8028, 26]-code), using
- discarding factors / shortening the dual code based on linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- discarding factors / shortening the dual code based on linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using
- OOA 12-folding and stacking with additional row [i] based on linear OA(2157, 8185, F2, 25) (dual of [8185, 8028, 26]-code), using
(157−25, 157, 1639)-Net over F2 — Digital
Digital (132, 157, 1639)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 1639, F2, 4, 25) (dual of [(1639, 4), 6399, 26]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 2048, F2, 4, 25) (dual of [(2048, 4), 8035, 26]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using
- an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- OOA 4-folding [i] based on linear OA(2157, 8192, F2, 25) (dual of [8192, 8035, 26]-code), using
- discarding factors / shortening the dual code based on linear OOA(2157, 2048, F2, 4, 25) (dual of [(2048, 4), 8035, 26]-NRT-code), using
(157−25, 157, 43308)-Net in Base 2 — Upper bound on s
There is no (132, 157, 43309)-net in base 2, because
- 1 times m-reduction [i] would yield (132, 156, 43309)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 91349 450750 406918 701825 053654 929259 562820 174037 > 2156 [i]