Best Known (162, 162+37, s)-Nets in Base 2
(162, 162+37, 272)-Net over F2 — Constructive and digital
Digital (162, 199, 272)-net over F2, using
- (u, u+v)-construction [i] based on
- digital (9, 27, 12)-net over F2, using
- net from sequence [i] based on digital (9, 11)-sequence over F2, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F2 with g(F) = 9 and N(F) ≥ 12, using
- Niederreiter–Xing sequence (Piršić implementation) with equidistant coordinate [i]
- net from sequence [i] based on digital (9, 11)-sequence over F2, using
- digital (135, 172, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 43, 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, 43, 65)-net over F16, using
- digital (9, 27, 12)-net over F2, using
(162, 162+37, 703)-Net over F2 — Digital
Digital (162, 199, 703)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2199, 703, F2, 2, 37) (dual of [(703, 2), 1207, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 1024, F2, 2, 37) (dual of [(1024, 2), 1849, 38]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- OOA 2-folding [i] based on linear OA(2199, 2048, F2, 37) (dual of [2048, 1849, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2199, 1024, F2, 2, 37) (dual of [(1024, 2), 1849, 38]-NRT-code), using
(162, 162+37, 15442)-Net in Base 2 — Upper bound on s
There is no (162, 199, 15443)-net in base 2, because
- 1 times m-reduction [i] would yield (162, 198, 15443)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 401995 819674 680550 870765 537996 929696 831135 839885 819900 060009 > 2198 [i]