Best Known (70−10, 70, s)-Nets in Base 2
(70−10, 70, 3276)-Net over F2 — Constructive and digital
Digital (60, 70, 3276)-net over F2, using
- net defined by OOA [i] based on linear OOA(270, 3276, F2, 10, 10) (dual of [(3276, 10), 32690, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(270, 16380, F2, 10) (dual of [16380, 16310, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(270, 16383, F2, 10) (dual of [16383, 16313, 11]-code), using
- 1 times truncation [i] based on linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 1 times truncation [i] based on linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(270, 16383, F2, 10) (dual of [16383, 16313, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(270, 16380, F2, 10) (dual of [16380, 16310, 11]-code), using
(70−10, 70, 5461)-Net over F2 — Digital
Digital (60, 70, 5461)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(270, 5461, F2, 3, 10) (dual of [(5461, 3), 16313, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(270, 16383, F2, 10) (dual of [16383, 16313, 11]-code), using
- 1 times truncation [i] based on linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 1 times truncation [i] based on linear OA(271, 16384, F2, 11) (dual of [16384, 16313, 12]-code), using
- OOA 3-folding [i] based on linear OA(270, 16383, F2, 10) (dual of [16383, 16313, 11]-code), using
(70−10, 70, 42676)-Net in Base 2 — Upper bound on s
There is no (60, 70, 42677)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 1180 712837 191352 968322 > 270 [i]