MinT - Best Known (164, 164+39, s)-Nets in Base 2

Best Known (164, 164+39, s)-Nets in Base 2

(164, 164+39, 268)-Net over F₂ — Constructive and digital

Digital (164, 203, 268)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (4, 23, 8)-net over F₂, using
  - net from sequence [i] based on digital (4, 7)-sequence over F₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 4 and N(F) ≥ 8, using
      - an explicitly constructive algebraic function field [i]
  - Niederreiterâ€“Xing sequence (PirÅ¡iÄ‡ implementation) with equidistant coordinate [i]
2. digital (141, 180, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 45, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(164, 164+39, 598)-Net over F₂ — Digital

Digital (164, 203, 598)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(164, 164+39, 12550)-Net in Base 2 — Upper bound on s

There is no (164, 203, 12551)-net in base 2, because

1 times m-reduction [i] would yield (164, 202, 12551)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 6 429289 932141 382084 388903 321666 392588 070028 591851 368095 404448 > 2²⁰² [i]