MinT - Best Known (140, 140+29, s)-Nets in Base 2

Best Known (140, 140+29, s)-Nets in Base 2

(140, 140+29, 320)-Net over F₂ — Constructive and digital

Digital (140, 169, 320)-net over F₂, using

t-expansion [i] based on digital (139, 169, 320)-net over F₂, using
- 1 times m-reduction [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
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 3 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]

(140, 140+29, 1024)-Net over F₂ — Digital

Digital (140, 169, 1024)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶⁹, 1024, F₂, 4, 29) (dual of [(1024, 4), 3927, 30]-NRT-code), using
- OOA 4-folding [i] based on linear OA(2¹⁶⁹, 4096, F₂, 29) (dual of [4096, 3927, 30]-code), using
  - an extension C_e(28) of the primitive narrow-sense BCH-code C(I) with length 4095Â = 2¹²âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]

(140, 140+29, 24743)-Net in Base 2 — Upper bound on s

There is no (140, 169, 24744)-net in base 2, because

1 times m-reduction [i] would yield (140, 168, 24744)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 374 286065 424554 819810 872647 063342 129161 611224 234108 > 2¹⁶⁸ [i]