Best Known (199, 199+28, s)-Nets in Base 2
(199, 199+28, 4682)-Net over F2 — Constructive and digital
Digital (199, 227, 4682)-net over F2, using
- 21 times duplication [i] based on digital (198, 226, 4682)-net over F2, using
- t-expansion [i] based on digital (197, 226, 4682)-net over F2, using
- net defined by OOA [i] based on linear OOA(2226, 4682, F2, 29, 29) (dual of [(4682, 29), 135552, 30]-NRT-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2226, 65549, F2, 29) (dual of [65549, 65323, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2226, 65553, F2, 29) (dual of [65553, 65327, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2225, 65536, F2, 29) (dual of [65536, 65311, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2226, 65553, F2, 29) (dual of [65553, 65327, 30]-code), using
- OOA 14-folding and stacking with additional row [i] based on linear OA(2226, 65549, F2, 29) (dual of [65549, 65323, 30]-code), using
- net defined by OOA [i] based on linear OOA(2226, 4682, F2, 29, 29) (dual of [(4682, 29), 135552, 30]-NRT-code), using
- t-expansion [i] based on digital (197, 226, 4682)-net over F2, using
(199, 199+28, 10925)-Net over F2 — Digital
Digital (199, 227, 10925)-net over F2, using
- 22 times duplication [i] based on digital (197, 225, 10925)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2225, 10925, F2, 6, 28) (dual of [(10925, 6), 65325, 29]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2225, 65550, F2, 28) (dual of [65550, 65325, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(2225, 65552, F2, 28) (dual of [65552, 65327, 29]-code), using
- 1 times truncation [i] based on linear OA(2226, 65553, F2, 29) (dual of [65553, 65327, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2225, 65536, F2, 29) (dual of [65536, 65311, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2209, 65536, F2, 27) (dual of [65536, 65327, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(21, 17, F2, 1) (dual of [17, 16, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- 1 times truncation [i] based on linear OA(2226, 65553, F2, 29) (dual of [65553, 65327, 30]-code), using
- discarding factors / shortening the dual code based on linear OA(2225, 65552, F2, 28) (dual of [65552, 65327, 29]-code), using
- OOA 6-folding [i] based on linear OA(2225, 65550, F2, 28) (dual of [65550, 65325, 29]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2225, 10925, F2, 6, 28) (dual of [(10925, 6), 65325, 29]-NRT-code), using
(199, 199+28, 459647)-Net in Base 2 — Upper bound on s
There is no (199, 227, 459648)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 215 684061 354770 439961 462194 992857 538307 811623 022037 570946 782099 391889 > 2227 [i]