Best Known (128, 128+17, s)-Nets in Base 2
(128, 128+17, 32768)-Net over F2 — Constructive and digital
Digital (128, 145, 32768)-net over F2, using
- net defined by OOA [i] based on linear OOA(2145, 32768, F2, 17, 17) (dual of [(32768, 17), 556911, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(2145, 262145, F2, 17) (dual of [262145, 262000, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 262145 | 236−1, 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(2145, 262145, F2, 17) (dual of [262145, 262000, 18]-code), using
(128, 128+17, 43690)-Net over F2 — Digital
Digital (128, 145, 43690)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2145, 43690, F2, 6, 17) (dual of [(43690, 6), 261995, 18]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2145, 262140, F2, 17) (dual of [262140, 261995, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(2145, 262144, F2, 17) (dual of [262144, 261999, 18]-code), using
- an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- discarding factors / shortening the dual code based on linear OA(2145, 262144, F2, 17) (dual of [262144, 261999, 18]-code), using
- OOA 6-folding [i] based on linear OA(2145, 262140, F2, 17) (dual of [262140, 261995, 18]-code), using
(128, 128+17, 986790)-Net in Base 2 — Upper bound on s
There is no (128, 145, 986791)-net in base 2, because
- 1 times m-reduction [i] would yield (128, 144, 986791)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 22 300849 404479 749583 568974 995808 341813 921700 > 2144 [i]