Best Known (189−32, 189, s)-Nets in Base 2
(189−32, 189, 380)-Net over F2 — Constructive and digital
Digital (157, 189, 380)-net over F2, using
- 1 times m-reduction [i] based on digital (157, 190, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 38, 76)-net over F32, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 5 and N(F) ≥ 76, using
- net from sequence [i] based on digital (5, 75)-sequence over F32, using
- trace code for nets [i] based on digital (5, 38, 76)-net over F32, using
(189−32, 189, 978)-Net over F2 — Digital
Digital (157, 189, 978)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2189, 978, F2, 2, 32) (dual of [(978, 2), 1767, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2189, 1046, F2, 2, 32) (dual of [(1046, 2), 1903, 33]-NRT-code), using
- strength reduction [i] based on linear OOA(2189, 1046, F2, 2, 33) (dual of [(1046, 2), 1903, 34]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2189, 2092, F2, 33) (dual of [2092, 1903, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2189, 2093, F2, 33) (dual of [2093, 1904, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- linear OA(2177, 2048, F2, 33) (dual of [2048, 1871, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2144, 2048, F2, 27) (dual of [2048, 1904, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(212, 45, F2, 5) (dual of [45, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- extracting embedded orthogonal array [i] based on digital (7, 11, 47)-net over F2, using
- adding a parity check bit [i] based on linear OA(211, 47, F2, 4) (dual of [47, 36, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(212, 48, F2, 5) (dual of [48, 36, 6]-code), using
- construction X applied to Ce(32) ⊂ Ce(26) [i] based on
- discarding factors / shortening the dual code based on linear OA(2189, 2093, F2, 33) (dual of [2093, 1904, 34]-code), using
- OOA 2-folding [i] based on linear OA(2189, 2092, F2, 33) (dual of [2092, 1903, 34]-code), using
- strength reduction [i] based on linear OOA(2189, 1046, F2, 2, 33) (dual of [(1046, 2), 1903, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2189, 1046, F2, 2, 32) (dual of [(1046, 2), 1903, 33]-NRT-code), using
(189−32, 189, 24436)-Net in Base 2 — Upper bound on s
There is no (157, 189, 24437)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 784 915275 669658 904263 941325 643864 742055 217144 602222 987599 > 2189 [i]