Best Known (212, 212+46, s)-Nets in Base 2
(212, 212+46, 380)-Net over F2 — Constructive and digital
Digital (212, 258, 380)-net over F2, using
- 23 times duplication [i] based on digital (209, 255, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 51, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 51, 76)-net over F32, using
(212, 212+46, 985)-Net over F2 — Digital
Digital (212, 258, 985)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2258, 985, F2, 2, 46) (dual of [(985, 2), 1712, 47]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2258, 1031, F2, 2, 46) (dual of [(1031, 2), 1804, 47]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2257, 1031, F2, 2, 46) (dual of [(1031, 2), 1805, 47]-NRT-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2255, 1030, F2, 2, 46) (dual of [(1030, 2), 1805, 47]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2255, 2060, F2, 46) (dual of [2060, 1805, 47]-code), using
- strength reduction [i] based on linear OA(2255, 2060, F2, 47) (dual of [2060, 1805, 48]-code), using
- construction X applied to Ce(46) ⊂ Ce(44) [i] based on
- linear OA(2254, 2048, F2, 47) (dual of [2048, 1794, 48]-code), using an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- linear OA(2243, 2048, F2, 45) (dual of [2048, 1805, 46]-code), using an extension Ce(44) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,44], and designed minimum distance d ≥ |I|+1 = 45 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 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(46) ⊂ Ce(44) [i] based on
- strength reduction [i] based on linear OA(2255, 2060, F2, 47) (dual of [2060, 1805, 48]-code), using
- OOA 2-folding [i] based on linear OA(2255, 2060, F2, 46) (dual of [2060, 1805, 47]-code), using
- 1 times NRT-code embedding in larger space [i] based on linear OOA(2255, 1030, F2, 2, 46) (dual of [(1030, 2), 1805, 47]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2257, 1031, F2, 2, 46) (dual of [(1031, 2), 1805, 47]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2258, 1031, F2, 2, 46) (dual of [(1031, 2), 1804, 47]-NRT-code), using
(212, 212+46, 22416)-Net in Base 2 — Upper bound on s
There is no (212, 258, 22417)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 463357 273351 524780 024104 897691 050940 822229 619738 976708 380681 189322 087500 145280 > 2258 [i]