Best Known (126−14, 126, s)-Nets in Base 2
(126−14, 126, 37449)-Net over F2 — Constructive and digital
Digital (112, 126, 37449)-net over F2, using
- net defined by OOA [i] based on linear OOA(2126, 37449, F2, 14, 14) (dual of [(37449, 14), 524160, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(2126, 262143, F2, 14) (dual of [262143, 262017, 15]-code), using
- 1 times truncation [i] based on linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 1 times truncation [i] based on linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using
- OA 7-folding and stacking [i] based on linear OA(2126, 262143, F2, 14) (dual of [262143, 262017, 15]-code), using
(126−14, 126, 52428)-Net over F2 — Digital
Digital (112, 126, 52428)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2126, 52428, F2, 5, 14) (dual of [(52428, 5), 262014, 15]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2126, 262140, F2, 14) (dual of [262140, 262014, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(2126, 262143, F2, 14) (dual of [262143, 262017, 15]-code), using
- 1 times truncation [i] based on linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using
- an extension Ce(14) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,14], and designed minimum distance d ≥ |I|+1 = 15 [i]
- 1 times truncation [i] based on linear OA(2127, 262144, F2, 15) (dual of [262144, 262017, 16]-code), using
- discarding factors / shortening the dual code based on linear OA(2126, 262143, F2, 14) (dual of [262143, 262017, 15]-code), using
- OOA 5-folding [i] based on linear OA(2126, 262140, F2, 14) (dual of [262140, 262014, 15]-code), using
(126−14, 126, 886040)-Net in Base 2 — Upper bound on s
There is no (112, 126, 886041)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 85 070797 434620 158108 377568 585430 594560 > 2126 [i]