Best Known (140, 171, s)-Nets in Base 2
(140, 171, 320)-Net over F2 — Constructive and digital
Digital (140, 171, 320)-net over F2, using
- 21 times duplication [i] based on digital (139, 170, 320)-net over F2, using
- trace code for nets [i] based on digital (3, 34, 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, 34, 64)-net over F32, using
(140, 171, 701)-Net over F2 — Digital
Digital (140, 171, 701)-net over F2, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2171, 701, F2, 2, 31) (dual of [(701, 2), 1231, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2171, 1032, F2, 2, 31) (dual of [(1032, 2), 1893, 32]-NRT-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2167, 1030, F2, 2, 31) (dual of [(1030, 2), 1893, 32]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2167, 2060, F2, 31) (dual of [2060, 1893, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(2166, 2048, F2, 31) (dual of [2048, 1882, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(2155, 2048, F2, 29) (dual of [2048, 1893, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 2047 = 211−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(21, 12, F2, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(21, s, F2, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- OOA 2-folding [i] based on linear OA(2167, 2060, F2, 31) (dual of [2060, 1893, 32]-code), using
- 2 times NRT-code embedding in larger space [i] based on linear OOA(2167, 1030, F2, 2, 31) (dual of [(1030, 2), 1893, 32]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2171, 1032, F2, 2, 31) (dual of [(1032, 2), 1893, 32]-NRT-code), using
(140, 171, 16552)-Net in Base 2 — Upper bound on s
There is no (140, 171, 16553)-net in base 2, because
- 1 times m-reduction [i] would yield (140, 170, 16553)-net in base 2, but
- the generalized Rao bound for nets shows that 2m ≥ 1497 263536 266983 875453 930282 292788 448833 005465 147008 > 2170 [i]