MinT - Best Known (123, 123+77, s)-Nets in Base 2

Best Known (123, 123+77, s)-Nets in Base 2

(123, 123+77, 68)-Net over F₂ — Constructive and digital

Digital (123, 200, 68)-net over F₂, using

4 times m-reduction [i] based on digital (123, 204, 68)-net over F₂, using
- trace code for nets [i] based on digital (21, 102, 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]

(123, 123+77, 101)-Net over F₂ — Digital

Digital (123, 200, 101)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+77, 512)-Net in Base 2 — Upper bound on s

There is no (123, 200, 513)-net in base 2, because

1 times m-reduction [i] would yield (123, 199, 513)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 830448 029800 014947 755989 363287 170387 861064 418860 632312 310528 > 2¹⁹⁹ [i]