Best Known (86, 86+10, s)-Nets in Base 2
(86, 86+10, 104861)-Net over F2 — Constructive and digital
Digital (86, 96, 104861)-net over F2, using
- net defined by OOA [i] based on linear OOA(296, 104861, F2, 10, 10) (dual of [(104861, 10), 1048514, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(296, 524305, F2, 10) (dual of [524305, 524209, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(296, 524308, F2, 10) (dual of [524308, 524212, 11]-code), using
- 1 times truncation [i] based on linear OA(297, 524309, F2, 11) (dual of [524309, 524212, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(296, 524288, F2, 11) (dual of [524288, 524192, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(277, 524288, F2, 9) (dual of [524288, 524211, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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
- 1 times truncation [i] based on linear OA(297, 524309, F2, 11) (dual of [524309, 524212, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(296, 524308, F2, 10) (dual of [524308, 524212, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(296, 524305, F2, 10) (dual of [524305, 524209, 11]-code), using
(86, 86+10, 138727)-Net over F2 — Digital
Digital (86, 96, 138727)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(296, 138727, F2, 3, 10) (dual of [(138727, 3), 416085, 11]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(296, 174769, F2, 3, 10) (dual of [(174769, 3), 524211, 11]-NRT-code), using
- OOA 3-folding [i] based on linear OA(296, 524307, F2, 10) (dual of [524307, 524211, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(296, 524308, F2, 10) (dual of [524308, 524212, 11]-code), using
- 1 times truncation [i] based on linear OA(297, 524309, F2, 11) (dual of [524309, 524212, 12]-code), using
- construction X4 applied to Ce(10) ⊂ Ce(8) [i] based on
- linear OA(296, 524288, F2, 11) (dual of [524288, 524192, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(277, 524288, F2, 9) (dual of [524288, 524211, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 524287 = 219−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(220, 21, F2, 19) (dual of [21, 1, 20]-code), using
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- dual of repetition code with length 21 [i]
- strength reduction [i] based on linear OA(220, 21, F2, 20) (dual of [21, 1, 21]-code or 21-arc in PG(19,2)), using
- linear OA(21, 21, F2, 1) (dual of [21, 20, 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
- 1 times truncation [i] based on linear OA(297, 524309, F2, 11) (dual of [524309, 524212, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(296, 524308, F2, 10) (dual of [524308, 524212, 11]-code), using
- OOA 3-folding [i] based on linear OA(296, 524307, F2, 10) (dual of [524307, 524211, 11]-code), using
- discarding factors / shortening the dual code based on linear OOA(296, 174769, F2, 3, 10) (dual of [(174769, 3), 524211, 11]-NRT-code), using
(86, 86+10, 1568954)-Net in Base 2 — Upper bound on s
There is no (86, 96, 1568955)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 79228 398954 454133 732804 129792 > 296 [i]