MinT - Best Known (239−91, 239, s)-Nets in Base 2

Best Known (239−91, 239, s)-Nets in Base 2

(239−91, 239, 77)-Net over F₂ — Constructive and digital

Digital (148, 239, 77)-net over F₂, using

1 times m-reduction [i] based on digital (148, 240, 77)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (48, 94, 35)-net over F₂, using
    - net from sequence [i] based on digital (48, 34)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 41, N(F)Â =Â 32, 1 place with degree 2, and 2 places with degree 4 [i] based on function field F/F₂ with g(F) = 41 and N(F) ≥ 32, using an explicitly constructive algebraic function field [i]
  2. digital (54, 146, 42)-net over F₂, using
    - net from sequence [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(239−91, 239, 84)-Net in Base 2 — Constructive

(148, 239, 84)-net in base 2, using

3 times m-reduction [i] based on (148, 242, 84)-net in base 2, using
- trace code for nets [i] based on (27, 121, 42)-net in base 4, using
  - net from sequence [i] based on (27, 41)-sequence in base 4, using
    - base expansion [i] based on digital (54, 41)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using
        an explicitly constructive algebraic function field [i]

(239−91, 239, 120)-Net over F₂ — Digital

Digital (148, 239, 120)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(239−91, 239, 624)-Net in Base 2 — Upper bound on s

There is no (148, 239, 625)-net in base 2, because

1 times m-reduction [i] would yield (148, 238, 625)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 443416 787477 994775 710215 719726 720164 085695 498481 474848 688686 836435 624776 > 2²³⁸ [i]