Best Known (211, 211+35, s)-Nets in Base 2
(211, 211+35, 965)-Net over F2 — Constructive and digital
Digital (211, 246, 965)-net over F2, using
- 21 times duplication [i] based on digital (210, 245, 965)-net over F2, using
- net defined by OOA [i] based on linear OOA(2245, 965, F2, 35, 35) (dual of [(965, 35), 33530, 36]-NRT-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2245, 16406, F2, 35) (dual of [16406, 16161, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2245, 16416, F2, 35) (dual of [16416, 16171, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2245, 16416, F2, 35) (dual of [16416, 16171, 36]-code), using
- OOA 17-folding and stacking with additional row [i] based on linear OA(2245, 16406, F2, 35) (dual of [16406, 16161, 36]-code), using
- net defined by OOA [i] based on linear OOA(2245, 965, F2, 35, 35) (dual of [(965, 35), 33530, 36]-NRT-code), using
(211, 211+35, 3283)-Net over F2 — Digital
Digital (211, 246, 3283)-net over F2, using
- 21 times duplication [i] based on digital (210, 245, 3283)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2245, 3283, F2, 5, 35) (dual of [(3283, 5), 16170, 36]-NRT-code), using
- OOA 5-folding [i] based on linear OA(2245, 16415, F2, 35) (dual of [16415, 16170, 36]-code), using
- discarding factors / shortening the dual code based on linear OA(2245, 16416, F2, 35) (dual of [16416, 16171, 36]-code), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- linear OA(2239, 16384, F2, 35) (dual of [16384, 16145, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(2211, 16384, F2, 31) (dual of [16384, 16173, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 16383 = 214−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(26, 32, F2, 3) (dual of [32, 26, 4]-code or 32-cap in PG(5,2)), using
- construction X applied to Ce(34) ⊂ Ce(30) [i] based on
- discarding factors / shortening the dual code based on linear OA(2245, 16416, F2, 35) (dual of [16416, 16171, 36]-code), using
- OOA 5-folding [i] based on linear OA(2245, 16415, F2, 35) (dual of [16415, 16170, 36]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2245, 3283, F2, 5, 35) (dual of [(3283, 5), 16170, 36]-NRT-code), using
(211, 211+35, 156404)-Net in Base 2 — Upper bound on s
There is no (211, 246, 156405)-net in base 2, because
- 1 times m-reduction [i] would yield (211, 245, 156405)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 56 542130 591352 944852 230146 170514 294842 223692 858409 287695 149887 611858 431178 > 2245 [i]