Best Known (112, 112+29, s)-Nets in Base 2
(112, 112+29, 260)-Net over F2 — Constructive and digital
Digital (112, 141, 260)-net over F2, using
- 21 times duplication [i] based on digital (111, 140, 260)-net over F2, using
- trace code for nets [i] based on digital (6, 35, 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, 35, 65)-net over F16, using
(112, 112+29, 393)-Net over F2 — Digital
Digital (112, 141, 393)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2141, 393, F2, 2, 29) (dual of [(393, 2), 645, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2141, 512, F2, 2, 29) (dual of [(512, 2), 883, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2141, 1024, F2, 29) (dual of [1024, 883, 30]-code), using
- an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 1023 = 210−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- OOA 2-folding [i] based on linear OA(2141, 1024, F2, 29) (dual of [1024, 883, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2141, 512, F2, 2, 29) (dual of [(512, 2), 883, 30]-NRT-code), using
(112, 112+29, 6170)-Net in Base 2 — Upper bound on s
There is no (112, 141, 6171)-net in base 2, because
- 1 times m-reduction [i] would yield (112, 140, 6171)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1 395416 423321 155720 882259 085828 481473 508836 > 2140 [i]