Best Known (198, 227, s)-Nets in Base 2
(198, 227, 4682)-Net over F2 — Constructive and digital
Digital (198, 227, 4682)-net over F2, using
- 21 times duplication [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
(198, 227, 9536)-Net over F2 — Digital
Digital (198, 227, 9536)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 9536, F2, 6, 29) (dual of [(9536, 6), 56989, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 10925, F2, 6, 29) (dual of [(10925, 6), 65323, 30]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2226, 10925, F2, 6, 29) (dual of [(10925, 6), 65324, 30]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2226, 65550, F2, 29) (dual of [65550, 65324, 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 6-folding [i] based on linear OA(2226, 65550, F2, 29) (dual of [65550, 65324, 30]-code), using
- 21 times duplication [i] based on linear OOA(2226, 10925, F2, 6, 29) (dual of [(10925, 6), 65324, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 10925, F2, 6, 29) (dual of [(10925, 6), 65323, 30]-NRT-code), using
(198, 227, 437443)-Net in Base 2 — Upper bound on s
There is no (198, 227, 437444)-net in base 2, because
- 1 times m-reduction [i] would yield (198, 226, 437444)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 107 842777 930313 658121 981479 805980 722607 663006 443763 040553 501061 770928 > 2226 [i]