MinT - Best Known (51, 51+25, s)-Nets in Base 2

Best Known (51, 51+25, s)-Nets in Base 2

(51, 51+25, 60)-Net over F₂ — Constructive and digital

Digital (51, 76, 60)-net over F₂, using

trace code for nets [i] based on digital (13, 38, 30)-net over F₄, using
- net from sequence [i] based on digital (13, 29)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
    - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(51, 51+25, 68)-Net over F₂ — Digital

Digital (51, 76, 68)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+25, 385)-Net in Base 2 — Upper bound on s

There is no (51, 76, 386)-net in base 2, because

1 times m-reduction [i] would yield (51, 75, 386)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 38500 385305 834562 967392 > 2⁷⁵ [i]