MinT - Best Known (138, 138+106, s)-Nets in Base 2

Best Known (138, 138+106, s)-Nets in Base 2

(138, 138+106, 67)-Net over F₂ — Constructive and digital

Digital (138, 244, 67)-net over F₂, using

2 times m-reduction [i] based on digital (138, 246, 67)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (39, 93, 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 (45, 153, 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]

(138, 138+106, 91)-Net over F₂ — Digital

Digital (138, 244, 91)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(138, 138+106, 427)-Net in Base 2 — Upper bound on s

There is no (138, 244, 428)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 31 274017 123844 125734 112697 649033 409715 681359 866936 869240 232928 701494 979824 > 2²⁴⁴ [i]