Best Known (254−47, 254, s)-Nets in Base 2
(254−47, 254, 320)-Net over F2 — Constructive and digital
Digital (207, 254, 320)-net over F2, using
- 1 times m-reduction [i] based on digital (207, 255, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 51, 64)-net over F32, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- net from sequence [i] based on digital (3, 63)-sequence over F32, using
- trace code for nets [i] based on digital (3, 51, 64)-net over F32, using
(254−47, 254, 852)-Net over F2 — Digital
Digital (207, 254, 852)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2254, 852, F2, 2, 47) (dual of [(852, 2), 1450, 48]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2254, 1024, F2, 2, 47) (dual of [(1024, 2), 1794, 48]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2254, 2048, F2, 47) (dual of [2048, 1794, 48]-code), using
- an extension Ce(46) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,46], and designed minimum distance d ≥ |I|+1 = 47 [i]
- OOA 2-folding [i] based on linear OA(2254, 2048, F2, 47) (dual of [2048, 1794, 48]-code), using
- discarding factors / shortening the dual code based on linear OOA(2254, 1024, F2, 2, 47) (dual of [(1024, 2), 1794, 48]-NRT-code), using
(254−47, 254, 19276)-Net in Base 2 — Upper bound on s
There is no (207, 254, 19277)-net in base 2, because
- 1 times m-reduction [i] would yield (207, 253, 19277)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 14487 752292 111626 379489 106737 438382 342146 409557 856677 066345 319177 233278 055200 > 2253 [i]