MinT - Best Known (144, 144+87, s)-Nets in Base 2

Best Known (144, 144+87, s)-Nets in Base 2

(144, 144+87, 76)-Net over F₂ — Constructive and digital

Digital (144, 231, 76)-net over F₂, using

3 times m-reduction [i] based on digital (144, 234, 76)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (45, 90, 34)-net over F₂, using
    - net from sequence [i] based on digital (45, 33)-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 1 place 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, 144, 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]

(144, 144+87, 84)-Net in Base 2 — Constructive

(144, 231, 84)-net in base 2, using

3 times m-reduction [i] based on (144, 234, 84)-net in base 2, using
- trace code for nets [i] based on (27, 117, 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]

(144, 144+87, 120)-Net over F₂ — Digital

Digital (144, 231, 120)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 144+87, 626)-Net in Base 2 — Upper bound on s

There is no (144, 231, 627)-net in base 2, because

1 times m-reduction [i] would yield (144, 230, 627)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1762 846438 068442 201067 827578 747781 460753 861886 792423 286643 576025 898180 > 2²³⁰ [i]