Best Known (82, 82+10, s)-Nets in Base 2
(82, 82+10, 52432)-Net over F2 — Constructive and digital
Digital (82, 92, 52432)-net over F2, using
- t-expansion [i] based on digital (81, 92, 52432)-net over F2, using
- net defined by OOA [i] based on linear OOA(292, 52432, F2, 11, 11) (dual of [(52432, 11), 576660, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(292, 262161, F2, 11) (dual of [262161, 262069, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(292, 262163, F2, 11) (dual of [262163, 262071, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(291, 262144, F2, 11) (dual of [262144, 262053, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(273, 262144, F2, 9) (dual of [262144, 262071, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(21, 19, F2, 1) (dual of [19, 18, 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(292, 262163, F2, 11) (dual of [262163, 262071, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(292, 262161, F2, 11) (dual of [262161, 262069, 12]-code), using
- net defined by OOA [i] based on linear OOA(292, 52432, F2, 11, 11) (dual of [(52432, 11), 576660, 12]-NRT-code), using
(82, 82+10, 87388)-Net over F2 — Digital
Digital (82, 92, 87388)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(292, 87388, F2, 3, 10) (dual of [(87388, 3), 262072, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(292, 262164, F2, 10) (dual of [262164, 262072, 11]-code), using
- strength reduction [i] based on linear OA(292, 262164, F2, 11) (dual of [262164, 262072, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(291, 262144, F2, 11) (dual of [262144, 262053, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(273, 262144, F2, 9) (dual of [262144, 262071, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(219, 20, F2, 19) (dual of [20, 1, 20]-code or 20-arc in PG(18,2)), using
- dual of repetition code with length 20 [i]
- linear OA(21, 20, F2, 1) (dual of [20, 19, 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 X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- strength reduction [i] based on linear OA(292, 262164, F2, 11) (dual of [262164, 262072, 12]-code), using
- OOA 3-folding [i] based on linear OA(292, 262164, F2, 10) (dual of [262164, 262072, 11]-code), using
(82, 82+10, 901124)-Net in Base 2 — Upper bound on s
There is no (82, 92, 901125)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 4951 773993 842917 196494 005226 > 292 [i]