MinT - Best Known (210−63, 210, s)-Nets in Base 2

Best Known (210−63, 210, s)-Nets in Base 2

(210−63, 210, 112)-Net over F₂ — Constructive and digital

Digital (147, 210, 112)-net over F₂, using

18 times m-reduction [i] based on digital (147, 228, 112)-net over F₂, using
- trace code for nets [i] based on digital (33, 114, 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]

(210−63, 210, 186)-Net over F₂ — Digital

Digital (147, 210, 186)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(210−63, 210, 1283)-Net in Base 2 — Upper bound on s

There is no (147, 210, 1284)-net in base 2, because

1 times m-reduction [i] would yield (147, 209, 1284)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 826 303911 687988 999573 541027 435452 156023 249568 828086 403073 643840 > 2²⁰⁹ [i]