Best Known (212, 212+37, s)-Nets in Base 2
(212, 212+37, 624)-Net over F2 — Constructive and digital
Digital (212, 249, 624)-net over F2, using
- 23 times duplication [i] based on digital (209, 246, 624)-net over F2, using
- t-expansion [i] based on digital (208, 246, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 41, 104)-net over F64, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F64 with g(F) = 3 and N(F) ≥ 104, using
- net from sequence [i] based on digital (3, 103)-sequence over F64, using
- trace code for nets [i] based on digital (3, 41, 104)-net over F64, using
- t-expansion [i] based on digital (208, 246, 624)-net over F2, using
(212, 212+37, 2261)-Net over F2 — Digital
Digital (212, 249, 2261)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2249, 2261, F2, 3, 37) (dual of [(2261, 3), 6534, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2249, 2748, F2, 3, 37) (dual of [(2748, 3), 7995, 38]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2248, 2748, F2, 3, 37) (dual of [(2748, 3), 7996, 38]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2248, 8244, F2, 37) (dual of [8244, 7996, 38]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- linear OA(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- linear OA(2196, 8192, F2, 31) (dual of [8192, 7996, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(213, 52, F2, 5) (dual of [52, 39, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(36) ⊂ Ce(30) [i] based on
- OOA 3-folding [i] based on linear OA(2248, 8244, F2, 37) (dual of [8244, 7996, 38]-code), using
- 21 times duplication [i] based on linear OOA(2248, 2748, F2, 3, 37) (dual of [(2748, 3), 7996, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2249, 2748, F2, 3, 37) (dual of [(2748, 3), 7995, 38]-NRT-code), using
(212, 212+37, 106058)-Net in Base 2 — Upper bound on s
There is no (212, 249, 106059)-net in base 2, because
- 1 times m-reduction [i] would yield (212, 248, 106059)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 452 342355 499888 445787 411331 366922 513721 581418 000855 568124 090112 390987 304660 > 2248 [i]