Best Known (86, 86+13, s)-Nets in Base 2
(86, 86+13, 10925)-Net over F2 — Constructive and digital
Digital (86, 99, 10925)-net over F2, using
- 21 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(298, 10925, F2, 13, 13) (dual of [(10925, 13), 141927, 14]-NRT-code), using
(86, 86+13, 14128)-Net over F2 — Digital
Digital (86, 99, 14128)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(299, 14128, F2, 4, 13) (dual of [(14128, 4), 56413, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(299, 16388, F2, 4, 13) (dual of [(16388, 4), 65453, 14]-NRT-code), using
- 21 times duplication [i] based on linear OOA(298, 16388, F2, 4, 13) (dual of [(16388, 4), 65454, 14]-NRT-code), using
- OOA 4-folding [i] based on linear OA(298, 65552, F2, 13) (dual of [65552, 65454, 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 4-folding [i] based on linear OA(298, 65552, F2, 13) (dual of [65552, 65454, 14]-code), using
- 21 times duplication [i] based on linear OOA(298, 16388, F2, 4, 13) (dual of [(16388, 4), 65454, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(299, 16388, F2, 4, 13) (dual of [(16388, 4), 65453, 14]-NRT-code), using
(86, 86+13, 247189)-Net in Base 2 — Upper bound on s
There is no (86, 99, 247190)-net in base 2, because
- 1 times m-reduction [i] would yield (86, 98, 247190)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 316914 776403 260191 902742 648304 > 298 [i]