MinT - Best Known (242−97, 242, s)-Nets in Base 2

Best Known (242−97, 242, s)-Nets in Base 2

(242−97, 242, 75)-Net over F₂ — Constructive and digital

Digital (145, 242, 75)-net over F₂, using

7 times m-reduction [i] based on digital (145, 249, 75)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 91, 33)-net over F₂, using
    - net from sequence [i] based on digital (39, 32)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 39 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (54, 158, 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]

(242−97, 242, 108)-Net over F₂ — Digital

Digital (145, 242, 108)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−97, 242, 540)-Net in Base 2 — Upper bound on s

There is no (145, 242, 541)-net in base 2, because

1 times m-reduction [i] would yield (145, 241, 541)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3 663279 366593 671191 732353 983832 019704 142523 362625 633687 903667 376321 170288 > 2²⁴¹ [i]