MinT - Best Known (219−80, 219, s)-Nets in Base 2

Best Known (219−80, 219, s)-Nets in Base 2

(219−80, 219, 76)-Net over F₂ — Constructive and digital

Digital (139, 219, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (45, 85, 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, 134, 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]

(219−80, 219, 84)-Net in Base 2 — Constructive

(139, 219, 84)-net in base 2, using

5 times m-reduction [i] based on (139, 224, 84)-net in base 2, using
- trace code for nets [i] based on (27, 112, 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]

(219−80, 219, 122)-Net over F₂ — Digital

Digital (139, 219, 122)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−80, 219, 644)-Net in Base 2 — Upper bound on s

There is no (139, 219, 645)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 889461 679766 677081 011133 635248 059300 463178 841506 969656 172466 167696 > 2²¹⁹ [i]