Best Known (55, 66, s)-Nets in Base 2
(55, 66, 1638)-Net over F2 — Constructive and digital
Digital (55, 66, 1638)-net over F2, using
- net defined by OOA [i] based on linear OOA(266, 1638, F2, 11, 11) (dual of [(1638, 11), 17952, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(266, 8191, F2, 11) (dual of [8191, 8125, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(266, 8191, F2, 11) (dual of [8191, 8125, 12]-code), using
(55, 66, 2048)-Net over F2 — Digital
Digital (55, 66, 2048)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(266, 2048, F2, 4, 11) (dual of [(2048, 4), 8126, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using
- an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- OOA 4-folding [i] based on linear OA(266, 8192, F2, 11) (dual of [8192, 8126, 12]-code), using
(55, 66, 21334)-Net in Base 2 — Upper bound on s
There is no (55, 66, 21335)-net in base 2, because
- 1 times m-reduction [i] would yield (55, 65, 21335)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 36 897269 783902 268268 > 265 [i]