Best Known (176−37, 176, s)-Nets in Base 2
(176−37, 176, 260)-Net over F2 — Constructive and digital
Digital (139, 176, 260)-net over F2, using
- t-expansion [i] based on digital (138, 176, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 44, 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, 44, 65)-net over F16, using
(176−37, 176, 423)-Net over F2 — Digital
Digital (139, 176, 423)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2176, 423, F2, 2, 37) (dual of [(423, 2), 670, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2176, 512, F2, 2, 37) (dual of [(512, 2), 848, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2176, 1024, F2, 37) (dual of [1024, 848, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- OOA 2-folding [i] based on linear OA(2176, 1024, F2, 37) (dual of [1024, 848, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2176, 512, F2, 2, 37) (dual of [(512, 2), 848, 38]-NRT-code), using
(176−37, 176, 6353)-Net in Base 2 — Upper bound on s
There is no (139, 176, 6354)-net in base 2, because
- 1 times m-reduction [i] would yield (139, 175, 6354)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 47971 424768 181372 149605 817672 599840 518188 760086 108104 > 2175 [i]