Best Known (65, 76, s)-Nets in Base 2
(65, 76, 6553)-Net over F2 — Constructive and digital
Digital (65, 76, 6553)-net over F2, using
- net defined by OOA [i] based on linear OOA(276, 6553, F2, 11, 11) (dual of [(6553, 11), 72007, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(276, 32766, F2, 11) (dual of [32766, 32690, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(276, 32766, F2, 11) (dual of [32766, 32690, 12]-code), using
(65, 76, 8192)-Net over F2 — Digital
Digital (65, 76, 8192)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(276, 8192, F2, 4, 11) (dual of [(8192, 4), 32692, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 215−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 4-folding [i] based on linear OA(276, 32768, F2, 11) (dual of [32768, 32692, 12]-code), using
(65, 76, 85359)-Net in Base 2 — Upper bound on s
There is no (65, 76, 85360)-net in base 2, because
- 1 times m-reduction [i] would yield (65, 75, 85360)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 37780 599537 274517 748573 > 275 [i]