Best Known (188, 227, s)-Nets in Base 2
(188, 227, 380)-Net over F2 — Constructive and digital
Digital (188, 227, 380)-net over F2, using
- 22 times duplication [i] based on digital (186, 225, 380)-net over F2, using
- t-expansion [i] based on digital (185, 225, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 45, 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, 45, 76)-net over F32, using
- t-expansion [i] based on digital (185, 225, 380)-net over F2, using
(188, 227, 1035)-Net over F2 — Digital
Digital (188, 227, 1035)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2227, 1035, F2, 2, 39) (dual of [(1035, 2), 1843, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 1049, F2, 2, 39) (dual of [(1049, 2), 1871, 40]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2223, 1047, F2, 2, 39) (dual of [(1047, 2), 1871, 40]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2223, 2094, F2, 39) (dual of [2094, 1871, 40]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- linear OA(2210, 2048, F2, 39) (dual of [2048, 1838, 40]-code), using an extension Ce(38) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,38], and designed minimum distance d ≥ |I|+1 = 39 [i]
- 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(213, 46, F2, 5) (dual of [46, 33, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 26−1, defining interval I = [0,3], and designed minimum distance d ≥ |I|+1 = 6 [i]
- discarding factors / shortening the dual code based on linear OA(213, 63, F2, 5) (dual of [63, 50, 6]-code), using
- construction X applied to Ce(38) ⊂ Ce(32) [i] based on
- OOA 2-folding [i] based on linear OA(2223, 2094, F2, 39) (dual of [2094, 1871, 40]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2223, 1047, F2, 2, 39) (dual of [(1047, 2), 1871, 40]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2227, 1049, F2, 2, 39) (dual of [(1049, 2), 1871, 40]-NRT-code), using
(188, 227, 30163)-Net in Base 2 — Upper bound on s
There is no (188, 227, 30164)-net in base 2, because
- 1 times m-reduction [i] would yield (188, 226, 30164)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 107 856897 207202 755502 730260 367968 475572 908982 654909 343004 364010 431851 > 2226 [i]