MinT - Best Known (211−69, 211, s)-Nets in Base 2

Best Known (211−69, 211, s)-Nets in Base 2

(211−69, 211, 112)-Net over F₂ — Constructive and digital

Digital (142, 211, 112)-net over F₂, using

7 times m-reduction [i] based on digital (142, 218, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 109, 56)-net over F₄, using
  - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
      - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(211−69, 211, 153)-Net over F₂ — Digital

Digital (142, 211, 153)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(211−69, 211, 929)-Net in Base 2 — Upper bound on s

There is no (142, 211, 930)-net in base 2, because

1 times m-reduction [i] would yield (142, 210, 930)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1665 658198 607098 244053 729391 244832 616282 521046 624539 700933 655528 > 2²¹⁰ [i]