Best Known (126, 126+18, s)-Nets in Base 2
(126, 126+18, 7281)-Net over F2 — Constructive and digital
Digital (126, 144, 7281)-net over F2, using
- net defined by OOA [i] based on linear OOA(2144, 7281, F2, 18, 18) (dual of [(7281, 18), 130914, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(2144, 65529, F2, 18) (dual of [65529, 65385, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(2144, 65535, F2, 18) (dual of [65535, 65391, 19]-code), using
- 1 times truncation [i] based on linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times truncation [i] based on linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using
- discarding factors / shortening the dual code based on linear OA(2144, 65535, F2, 18) (dual of [65535, 65391, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(2144, 65529, F2, 18) (dual of [65529, 65385, 19]-code), using
(126, 126+18, 13107)-Net over F2 — Digital
Digital (126, 144, 13107)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2144, 13107, F2, 5, 18) (dual of [(13107, 5), 65391, 19]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2144, 65535, F2, 18) (dual of [65535, 65391, 19]-code), using
- 1 times truncation [i] based on linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using
- an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- 1 times truncation [i] based on linear OA(2145, 65536, F2, 19) (dual of [65536, 65391, 20]-code), using
- OOA 5-folding [i] based on linear OA(2144, 65535, F2, 18) (dual of [65535, 65391, 19]-code), using
(126, 126+18, 271775)-Net in Base 2 — Upper bound on s
There is no (126, 144, 271776)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 22 300974 688195 888357 527194 969127 337851 214117 > 2144 [i]