MinT - Best Known (172−57, 172, s)-Nets in Base 2

Best Known (172−57, 172, s)-Nets in Base 2

(172−57, 172, 69)-Net over F₂ — Constructive and digital

Digital (115, 172, 69)-net over F₂, using

2¹ times duplication [i] based on digital (114, 171, 69)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (38, 66, 34)-net over F₂, using
    - trace code for nets [i] based on digital (5, 33, 17)-net over F₄, using
      - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (48, 105, 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]

(172−57, 172, 84)-Net in Base 2 — Constructive

(115, 172, 84)-net in base 2, using

4 times m-reduction [i] based on (115, 176, 84)-net in base 2, using
- trace code for nets [i] based on (27, 88, 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]

(172−57, 172, 125)-Net over F₂ — Digital

Digital (115, 172, 125)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(172−57, 172, 738)-Net in Base 2 — Upper bound on s

There is no (115, 172, 739)-net in base 2, because

1 times m-reduction [i] would yield (115, 171, 739)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 3069 075461 726765 177562 994165 680131 720706 658234 113208 > 2¹⁷¹ [i]