Best Known (118, 118+27, s)-Nets in Base 2
(118, 118+27, 263)-Net over F2 — Constructive and digital
Digital (118, 145, 263)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (0, 13, 3)-net over F2, using
- net from sequence [i] based on digital (0, 2)-sequence over F2, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 0 and N(F) ≥ 3, using
- the rational function field F2(x) [i]
- Niederreiter sequence [i]
- Sobol sequence [i]
- net from sequence [i] based on digital (0, 2)-sequence over F2, using
- digital (105, 132, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 33, 65)-net over F16, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- the Hermitian function field over F16 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F16 with g(F) = 6 and N(F) ≥ 65, using
- net from sequence [i] based on digital (6, 64)-sequence over F16, using
- trace code for nets [i] based on digital (6, 33, 65)-net over F16, using
- digital (0, 13, 3)-net over F2, using
(118, 118+27, 648)-Net over F2 — Digital
Digital (118, 145, 648)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2145, 648, F2, 3, 27) (dual of [(648, 3), 1799, 28]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2145, 686, F2, 3, 27) (dual of [(686, 3), 1913, 28]-NRT-code), using
- OOA 3-folding [i] based on linear OA(2145, 2058, F2, 27) (dual of [2058, 1913, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(2145, 2060, F2, 27) (dual of [2060, 1915, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- 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(2133, 2048, F2, 25) (dual of [2048, 1915, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(2145, 2060, F2, 27) (dual of [2060, 1915, 28]-code), using
- OOA 3-folding [i] based on linear OA(2145, 2058, F2, 27) (dual of [2058, 1913, 28]-code), using
- discarding factors / shortening the dual code based on linear OOA(2145, 686, F2, 3, 27) (dual of [(686, 3), 1913, 28]-NRT-code), using
(118, 118+27, 12224)-Net in Base 2 — Upper bound on s
There is no (118, 145, 12225)-net in base 2, because
- 1 times m-reduction [i] would yield (118, 144, 12225)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 22 321158 035159 520285 387429 622587 969048 111776 > 2144 [i]