Best Known (205−31, 205, s)-Nets in Base 2
(205−31, 205, 624)-Net over F2 — Constructive and digital
Digital (174, 205, 624)-net over F2, using
- 21 times duplication [i] based on digital (173, 204, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 34, 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, 34, 104)-net over F64, using
(205−31, 205, 2056)-Net over F2 — Digital
Digital (174, 205, 2056)-net over F2, using
- 22 times duplication [i] based on digital (172, 203, 2056)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2203, 2056, F2, 4, 31) (dual of [(2056, 4), 8021, 32]-NRT-code), using
- 21 times duplication [i] based on linear OOA(2202, 2056, F2, 4, 31) (dual of [(2056, 4), 8022, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2202, 8224, F2, 31) (dual of [8224, 8022, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(26) [i] based on
- 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(2170, 8192, F2, 27) (dual of [8192, 8022, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [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(30) ⊂ Ce(26) [i] based on
- OOA 4-folding [i] based on linear OA(2202, 8224, F2, 31) (dual of [8224, 8022, 32]-code), using
- 21 times duplication [i] based on linear OOA(2202, 2056, F2, 4, 31) (dual of [(2056, 4), 8022, 32]-NRT-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2203, 2056, F2, 4, 31) (dual of [(2056, 4), 8021, 32]-NRT-code), using
(205−31, 205, 79736)-Net in Base 2 — Upper bound on s
There is no (174, 205, 79737)-net in base 2, because
- 1 times m-reduction [i] would yield (174, 204, 79737)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 25 715541 161428 112031 248026 040487 991159 100090 130000 436251 847168 > 2204 [i]