Best Known (71, 82, s)-Nets in Base 2
(71, 82, 13110)-Net over F2 — Constructive and digital
Digital (71, 82, 13110)-net over F2, using
- net defined by OOA [i] based on linear OOA(282, 13110, F2, 11, 11) (dual of [(13110, 11), 144128, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(282, 65551, F2, 11) (dual of [65551, 65469, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(282, 65553, F2, 11) (dual of [65553, 65471, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- 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(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(282, 65553, F2, 11) (dual of [65553, 65471, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(282, 65551, F2, 11) (dual of [65551, 65469, 12]-code), using
(71, 82, 16388)-Net over F2 — Digital
Digital (71, 82, 16388)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(282, 16388, F2, 4, 11) (dual of [(16388, 4), 65470, 12]-NRT-code), using
- OOA 4-folding [i] based on linear OA(282, 65552, F2, 11) (dual of [65552, 65470, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(282, 65553, F2, 11) (dual of [65553, 65471, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- 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(265, 65536, F2, 9) (dual of [65536, 65471, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 216−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [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(10) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(282, 65553, F2, 11) (dual of [65553, 65471, 12]-code), using
- OOA 4-folding [i] based on linear OA(282, 65552, F2, 11) (dual of [65552, 65470, 12]-code), using
(71, 82, 196113)-Net in Base 2 — Upper bound on s
There is no (71, 82, 196114)-net in base 2, because
- 1 times m-reduction [i] would yield (71, 81, 196114)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 2 417905 080358 032172 252884 > 281 [i]