MinT - Best Known (123−51, 123, s)-Nets in Base 2

Best Known (123−51, 123, s)-Nets in Base 2

(123−51, 123, 54)-Net over F₂ — Constructive and digital

Digital (72, 123, 54)-net over F₂, using

1 times m-reduction [i] based on digital (72, 124, 54)-net over F₂, using
- trace code for nets [i] based on digital (10, 62, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]

(123−51, 123, 57)-Net over F₂ — Digital

Digital (72, 123, 57)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−51, 123, 264)-Net in Base 2 — Upper bound on s

There is no (72, 123, 265)-net in base 2, because

1 times m-reduction [i] would yield (72, 122, 265)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 5 485500 460951 820356 043962 417317 939232 > 2¹²² [i]