Best Known (206−43, 206, s)-Nets in Base 2
(206−43, 206, 260)-Net over F2 — Constructive and digital
Digital (163, 206, 260)-net over F2, using
- t-expansion [i] based on digital (162, 206, 260)-net over F2, using
- 2 times m-reduction [i] based on digital (162, 208, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 52, 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, 52, 65)-net over F16, using
- 2 times m-reduction [i] based on digital (162, 208, 260)-net over F2, using
(206−43, 206, 486)-Net over F2 — Digital
Digital (163, 206, 486)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2206, 486, F2, 2, 43) (dual of [(486, 2), 766, 44]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 512, F2, 2, 43) (dual of [(512, 2), 818, 44]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2206, 1024, F2, 43) (dual of [1024, 818, 44]-code), using
- an extension Ce(42) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,42], and designed minimum distance d ≥ |I|+1 = 43 [i]
- OOA 2-folding [i] based on linear OA(2206, 1024, F2, 43) (dual of [1024, 818, 44]-code), using
- discarding factors / shortening the dual code based on linear OOA(2206, 512, F2, 2, 43) (dual of [(512, 2), 818, 44]-NRT-code), using
(206−43, 206, 7504)-Net in Base 2 — Upper bound on s
There is no (163, 206, 7505)-net in base 2, because
- 1 times m-reduction [i] would yield (163, 205, 7505)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 51 468860 107942 223735 532903 997573 379697 622897 844262 004157 540416 > 2205 [i]