Best Known (216, 216+39, s)-Nets in Base 2
(216, 216+39, 624)-Net over F2 — Constructive and digital
Digital (216, 255, 624)-net over F2, using
- 23 times duplication [i] based on digital (213, 252, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 42, 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, 42, 104)-net over F64, using
(216, 216+39, 2056)-Net over F2 — Digital
Digital (216, 255, 2056)-net over F2, using
- 21 times duplication [i] based on digital (215, 254, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 2056, F2, 4, 39) (dual of [(2056, 4), 7970, 40]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2254, 8224, F2, 39) (dual of [8224, 7970, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(34) [i] based on
- linear OA(2248, 8192, F2, 39) (dual of [8192, 7944, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2222, 8192, F2, 35) (dual of [8192, 7970, 36]-code), using an extension Ce(34) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,34], and designed minimum distance d ≥ |I|+1 = 35 [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(38) ⊂ Ce(34) [i] based on
- OOA 4-folding [i] based on linear OA(2254, 8224, F2, 39) (dual of [8224, 7970, 40]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 2056, F2, 4, 39) (dual of [(2056, 4), 7970, 40]-NRT-code), using
(216, 216+39, 83823)-Net in Base 2 — Upper bound on s
There is no (216, 255, 83824)-net in base 2, because
- 1 times m-reduction [i] would yield (216, 254, 83824)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 28950 765225 227977 405339 416036 838917 305324 668094 513673 297840 476722 894571 427502 > 2254 [i]