MinT - Best Known (213−89, 213, s)-Nets in Base 2

Best Known (213−89, 213, s)-Nets in Base 2

(213−89, 213, 66)-Net over F₂ — Constructive and digital

Digital (124, 213, 66)-net over F₂, using

5 times m-reduction [i] based on digital (124, 218, 66)-net over F₂, using
- trace code for nets [i] based on digital (15, 109, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]

(213−89, 213, 88)-Net over F₂ — Digital

Digital (124, 213, 88)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(213−89, 213, 425)-Net in Base 2 — Upper bound on s

There is no (124, 213, 426)-net in base 2, because

1 times m-reduction [i] would yield (124, 212, 426)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7179 292521 584807 585494 875628 072009 069171 113874 252809 954532 860272 > 2²¹² [i]