MinT - Best Known (190−68, 190, s)-Nets in Base 2

Best Known (190−68, 190, s)-Nets in Base 2

(190−68, 190, 68)-Net over F₂ — Constructive and digital

Digital (122, 190, 68)-net over F₂, using

12 times m-reduction [i] based on digital (122, 202, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 101, 34)-net over F₄, using
  - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
      - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(190−68, 190, 84)-Net in Base 2 — Constructive

(122, 190, 84)-net in base 2, using

trace code for nets [i] based on (27, 95, 42)-net in base 4, using
- net from sequence [i] based on (27, 41)-sequence in base 4, using
  - base expansion [i] based on digital (54, 41)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
      - an explicitly constructive algebraic function field [i]

(190−68, 190, 114)-Net over F₂ — Digital

Digital (122, 190, 114)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−68, 190, 602)-Net in Base 2 — Upper bound on s

There is no (122, 190, 603)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1619 354936 004886 816668 248490 203127 436676 839040 948449 081578 > 2¹⁹⁰ [i]