MinT - Best Known (89, 89+47, s)-Nets in Base 2

Best Known (89, 89+47, s)-Nets in Base 2

(89, 89+47, 68)-Net over F₂ — Constructive and digital

Digital (89, 136, 68)-net over F₂, using

trace code for nets [i] based on digital (21, 68, 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]

(89, 89+47, 94)-Net over F₂ — Digital

Digital (89, 136, 94)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+47, 518)-Net in Base 2 — Upper bound on s

There is no (89, 136, 519)-net in base 2, because

1 times m-reduction [i] would yield (89, 135, 519)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 45028 613536 967511 000009 585942 712548 230976 > 2¹³⁵ [i]