Best Known (188−32, 188, s)-Nets in Base 2
(188−32, 188, 380)-Net over F2 — Constructive and digital
Digital (156, 188, 380)-net over F2, using
- 23 times duplication [i] based on digital (153, 185, 380)-net over F2, using
- trace code for nets [i] based on digital (5, 37, 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, 37, 76)-net over F32, using
(188−32, 188, 954)-Net over F2 — Digital
Digital (156, 188, 954)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2188, 954, F2, 2, 32) (dual of [(954, 2), 1720, 33]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2188, 1046, F2, 2, 32) (dual of [(1046, 2), 1904, 33]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2188, 2092, F2, 32) (dual of [2092, 1904, 33]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(2189, 2093, F2, 33) (dual of [2093, 1904, 34]-code), using
- OOA 2-folding [i] based on linear OA(2188, 2092, F2, 32) (dual of [2092, 1904, 33]-code), using
- discarding factors / shortening the dual code based on linear OOA(2188, 1046, F2, 2, 32) (dual of [(1046, 2), 1904, 33]-NRT-code), using
(188−32, 188, 23399)-Net in Base 2 — Upper bound on s
There is no (156, 188, 23400)-net in base 2, because
- the generalized Rao bound for nets shows that 2m ≥ 392 469450 935055 038365 124275 064394 625894 719501 341836 047546 > 2188 [i]