Best Known (85, 85+13, s)-Nets in Base 2
(85, 85+13, 10925)-Net over F2 — Constructive and digital
Digital (85, 98, 10925)-net over F2, using
- net defined by OOA [i] based on linear OOA(298, 10925, F2, 13, 13) (dual of [(10925, 13), 141927, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(298, 65551, F2, 13) (dual of [65551, 65453, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(298, 65553, F2, 13) (dual of [65553, 65455, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(297, 65536, F2, 13) (dual of [65536, 65439, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(281, 65536, F2, 11) (dual of [65536, 65455, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(298, 65553, F2, 13) (dual of [65553, 65455, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(298, 65551, F2, 13) (dual of [65551, 65453, 14]-code), using
(85, 85+13, 13110)-Net over F2 — Digital
Digital (85, 98, 13110)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(298, 13110, F2, 5, 13) (dual of [(13110, 5), 65452, 14]-NRT-code), using
- OOA 5-folding [i] based on linear OA(298, 65550, F2, 13) (dual of [65550, 65452, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(298, 65553, F2, 13) (dual of [65553, 65455, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(297, 65536, F2, 13) (dual of [65536, 65439, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(281, 65536, F2, 11) (dual of [65536, 65455, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(12) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(298, 65553, F2, 13) (dual of [65553, 65455, 14]-code), using
- OOA 5-folding [i] based on linear OA(298, 65550, F2, 13) (dual of [65550, 65452, 14]-code), using
(85, 85+13, 220220)-Net in Base 2 — Upper bound on s
There is no (85, 98, 220221)-net in base 2, because
- 1 times m-reduction [i] would yield (85, 97, 220221)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 158460 294454 099620 050901 961192 > 297 [i]