MinT - Best Known (109−45, 109, s)-Nets in Base 2

Best Known (109−45, 109, s)-Nets in Base 2

(109−45, 109, 44)-Net over F₂ — Constructive and digital

Digital (64, 109, 44)-net over F₂, using

1 times m-reduction [i] based on digital (64, 110, 44)-net over F₂, using
- trace code for nets [i] based on digital (9, 55, 22)-net over F₄, using
  - net from sequence [i] based on digital (9, 21)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 9 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]

(109−45, 109, 53)-Net over F₂ — Digital

Digital (64, 109, 53)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−45, 109, 241)-Net in Base 2 — Upper bound on s

There is no (64, 109, 242)-net in base 2, because

1 times m-reduction [i] would yield (64, 108, 242)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 346 228430 699607 207817 663575 729456 > 2¹⁰⁸ [i]