MinT - Best Known (68−19, 68, s)-Nets in Base 2

Best Known (68−19, 68, s)-Nets in Base 2

(68−19, 68, 72)-Net over F₂ — Constructive and digital

Digital (49, 68, 72)-net over F₂, using

1 times m-reduction [i] based on digital (49, 69, 72)-net over F₂, using
- trace code for nets [i] based on digital (3, 23, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]

(68−19, 68, 98)-Net over F₂ — Digital

Digital (49, 68, 98)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2⁶⁸, 98, F₂, 2, 19) (dual of [(98, 2), 128, 20]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2⁶⁸, 196, F₂, 19) (dual of [196, 128, 20]-code), using
  - a code from Grasslâ€™s database [i]

(68−19, 68, 709)-Net in Base 2 — Upper bound on s

There is no (49, 68, 710)-net in base 2, because

1 times m-reduction [i] would yield (49, 67, 710)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 148 567345 379580 561128 > 2⁶⁷ [i]