MinT - Best Known (242−100, 242, s)-Nets in Base 2

Best Known (242−100, 242, s)-Nets in Base 2

(242−100, 242, 70)-Net over F₂ — Constructive and digital

Digital (142, 242, 70)-net over F₂, using

2 times m-reduction [i] based on digital (142, 244, 70)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 72, 21)-net over F₂, using
    - net from sequence [i] based on digital (21, 20)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 21 and N(F) ≥ 21, using
        an explicitly constructive algebraic function field [i]
  2. digital (70, 172, 49)-net over F₂, using
    - net from sequence [i] based on digital (70, 48)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, and 1 place with degree 2 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(242−100, 242, 101)-Net over F₂ — Digital

Digital (142, 242, 101)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(242−100, 242, 487)-Net in Base 2 — Upper bound on s

There is no (142, 242, 488)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7 221858 785671 576803 455501 907163 777136 378088 681129 956936 911228 904297 981876 > 2²⁴² [i]