Best Known (157−27, 157, s)-Nets in Base 2
(157−27, 157, 320)-Net over F2 — Constructive and digital
Digital (130, 157, 320)-net over F2, using
- 22 times duplication [i] based on digital (128, 155, 320)-net over F2, using
- t-expansion [i] based on digital (127, 155, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 31, 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, 31, 64)-net over F32, using
- t-expansion [i] based on digital (127, 155, 320)-net over F2, using
(157−27, 157, 1024)-Net over F2 — Digital
Digital (130, 157, 1024)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2157, 1024, F2, 4, 27) (dual of [(1024, 4), 3939, 28]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using
- an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 4095 = 212−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- OOA 4-folding [i] based on linear OA(2157, 4096, F2, 27) (dual of [4096, 3939, 28]-code), using
(157−27, 157, 23195)-Net in Base 2 — Upper bound on s
There is no (130, 157, 23196)-net in base 2, because
- 1 times m-reduction [i] would yield (130, 156, 23196)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 91350 498332 337409 435960 257095 568045 415707 786988 > 2156 [i]