Best Known (164, 164+29, s)-Nets in Base 2
(164, 164+29, 624)-Net over F2 — Constructive and digital
Digital (164, 193, 624)-net over F2, using
- 21 times duplication [i] based on digital (163, 192, 624)-net over F2, using
- trace code for nets [i] based on digital (3, 32, 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, 32, 104)-net over F64, using
(164, 164+29, 2057)-Net over F2 — Digital
Digital (164, 193, 2057)-net over F2, using
- 21 times duplication [i] based on digital (163, 192, 2057)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2192, 2057, F2, 4, 29) (dual of [(2057, 4), 8036, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2192, 8228, F2, 29) (dual of [8228, 8036, 30]-code), using
- 3 times code embedding in larger space [i] based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- construction X applied to C([0,14]) ⊂ C([0,12]) [i] based on
- linear OA(2183, 8193, F2, 29) (dual of [8193, 8010, 30]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,14], and minimum distance d ≥ |{−14,−13,…,14}|+1 = 30 (BCH-bound) [i]
- linear OA(2157, 8193, F2, 25) (dual of [8193, 8036, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 8193 | 226−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [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 C([0,14]) ⊂ C([0,12]) [i] based on
- 3 times code embedding in larger space [i] based on linear OA(2189, 8225, F2, 29) (dual of [8225, 8036, 30]-code), using
- OOA 4-folding [i] based on linear OA(2192, 8228, F2, 29) (dual of [8228, 8036, 30]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2192, 2057, F2, 4, 29) (dual of [(2057, 4), 8036, 30]-NRT-code), using
(164, 164+29, 81238)-Net in Base 2 — Upper bound on s
There is no (164, 193, 81239)-net in base 2, because
- 1 times m-reduction [i] would yield (164, 192, 81239)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 6278 118562 450915 708383 421691 190657 717181 827533 611890 717306 > 2192 [i]