Best Known (235−37, 235, s)-Nets in Base 2
(235−37, 235, 520)-Net over F2 — Constructive and digital
Digital (198, 235, 520)-net over F2, using
- t-expansion [i] based on digital (197, 235, 520)-net over F2, using
- trace code for nets [i] based on digital (9, 47, 104)-net over F32, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F32, using
- trace code for nets [i] based on digital (9, 47, 104)-net over F32, using
(235−37, 235, 1859)-Net over F2 — Digital
Digital (198, 235, 1859)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2235, 1859, F2, 4, 37) (dual of [(1859, 4), 7201, 38]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2235, 2048, F2, 4, 37) (dual of [(2048, 4), 7957, 38]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using
- an extension Ce(36) of the primitive narrow-sense BCH-code C(I) with length 8191 = 213−1, defining interval I = [1,36], and designed minimum distance d ≥ |I|+1 = 37 [i]
- OOA 4-folding [i] based on linear OA(2235, 8192, F2, 37) (dual of [8192, 7957, 38]-code), using
- discarding factors / shortening the dual code based on linear OOA(2235, 2048, F2, 4, 37) (dual of [(2048, 4), 7957, 38]-NRT-code), using
(235−37, 235, 61849)-Net in Base 2 — Upper bound on s
There is no (198, 235, 61850)-net in base 2, because
- 1 times m-reduction [i] would yield (198, 234, 61850)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 27613 512324 511284 119238 130919 020794 257993 622561 460945 906635 526589 407156 > 2234 [i]