Best Known (212, 212+27, s)-Nets in Base 2
(212, 212+27, 20166)-Net over F2 — Constructive and digital
Digital (212, 239, 20166)-net over F2, using
- 23 times duplication [i] based on digital (209, 236, 20166)-net over F2, using
- net defined by OOA [i] based on linear OOA(2236, 20166, F2, 27, 27) (dual of [(20166, 27), 544246, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2236, 262159, F2, 27) (dual of [262159, 261923, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2236, 262163, F2, 27) (dual of [262163, 261927, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2217, 262144, F2, 25) (dual of [262144, 261927, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2236, 262163, F2, 27) (dual of [262163, 261927, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(2236, 262159, F2, 27) (dual of [262159, 261923, 28]-code), using
- net defined by OOA [i] based on linear OOA(2236, 20166, F2, 27, 27) (dual of [(20166, 27), 544246, 28]-NRT-code), using
(212, 212+27, 37452)-Net over F2 — Digital
Digital (212, 239, 37452)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2239, 37452, F2, 7, 27) (dual of [(37452, 7), 261925, 28]-NRT-code), using
- 22 times duplication [i] based on linear OOA(2237, 37452, F2, 7, 27) (dual of [(37452, 7), 261927, 28]-NRT-code), using
- OOA 7-folding [i] based on linear OA(2237, 262164, F2, 27) (dual of [262164, 261927, 28]-code), using
- 1 times code embedding in larger space [i] based on linear OA(2236, 262163, F2, 27) (dual of [262163, 261927, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(2235, 262144, F2, 27) (dual of [262144, 261909, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(2217, 262144, F2, 25) (dual of [262144, 261927, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 262143 = 218−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [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(26) ⊂ Ce(24) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(2236, 262163, F2, 27) (dual of [262163, 261927, 28]-code), using
- OOA 7-folding [i] based on linear OA(2237, 262164, F2, 27) (dual of [262164, 261927, 28]-code), using
- 22 times duplication [i] based on linear OOA(2237, 37452, F2, 7, 27) (dual of [(37452, 7), 261927, 28]-NRT-code), using
(212, 212+27, 1838934)-Net in Base 2 — Upper bound on s
There is no (212, 239, 1838935)-net in base 2, because
- 1 times m-reduction [i] would yield (212, 238, 1838935)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 441713 771046 467069 825376 526396 526292 903446 576104 514444 253578 571706 125448 > 2238 [i]