MinT - Best Known (226−85, 226, s)-Nets in Base 2

Best Known (226−85, 226, s)-Nets in Base 2

(226−85, 226, 76)-Net over F₂ — Constructive and digital

Digital (141, 226, 76)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (45, 87, 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, 139, 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]

(226−85, 226, 84)-Net in Base 2 — Constructive

(141, 226, 84)-net in base 2, using

2 times m-reduction [i] based on (141, 228, 84)-net in base 2, using
- trace code for nets [i] based on (27, 114, 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]

(226−85, 226, 118)-Net over F₂ — Digital

Digital (141, 226, 118)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−85, 226, 616)-Net in Base 2 — Upper bound on s

There is no (141, 226, 617)-net in base 2, because

1 times m-reduction [i] would yield (141, 225, 617)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 54 531914 749531 509427 972036 586895 042923 372202 525330 195322 963770 317004 > 2²²⁵ [i]