Best Known (165, 165+21, s)-Nets in Base 2
(165, 165+21, 26216)-Net over F2 — Constructive and digital
Digital (165, 186, 26216)-net over F2, using
- 24 times duplication [i] based on digital (161, 182, 26216)-net over F2, using
- net defined by OOA [i] based on linear OOA(2182, 26216, F2, 21, 21) (dual of [(26216, 21), 550354, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2182, 262161, F2, 21) (dual of [262161, 261979, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(2182, 262163, F2, 21) (dual of [262163, 261981, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2163, 262144, F2, 19) (dual of [262144, 261981, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(2182, 262163, F2, 21) (dual of [262163, 261981, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(2182, 262161, F2, 21) (dual of [262161, 261979, 22]-code), using
- net defined by OOA [i] based on linear OOA(2182, 26216, F2, 21, 21) (dual of [(26216, 21), 550354, 22]-NRT-code), using
(165, 165+21, 43694)-Net over F2 — Digital
Digital (165, 186, 43694)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2186, 43694, F2, 6, 21) (dual of [(43694, 6), 261978, 22]-NRT-code), using
- 23 times duplication [i] based on linear OOA(2183, 43694, F2, 6, 21) (dual of [(43694, 6), 261981, 22]-NRT-code), using
- OOA 6-folding [i] based on linear OA(2183, 262164, F2, 21) (dual of [262164, 261981, 22]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2182, 262163, F2, 21) (dual of [262163, 261981, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(18) [i] based on
- linear OA(2181, 262144, F2, 21) (dual of [262144, 261963, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(2163, 262144, F2, 19) (dual of [262144, 261981, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(20) ⊂ Ce(18) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2182, 262163, F2, 21) (dual of [262163, 261981, 22]-code), using
- OOA 6-folding [i] based on linear OA(2183, 262164, F2, 21) (dual of [262164, 261981, 22]-code), using
- 23 times duplication [i] based on linear OOA(2183, 43694, F2, 6, 21) (dual of [(43694, 6), 261981, 22]-NRT-code), using
(165, 165+21, 1678910)-Net in Base 2 — Upper bound on s
There is no (165, 186, 1678911)-net in base 2, because
- 1 times m-reduction [i] would yield (165, 185, 1678911)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 49 040085 699362 877803 654943 896749 452669 742512 925735 637743 > 2185 [i]