MinT - Best Known (237−89, 237, s)-Nets in Base 2

Best Known (237−89, 237, s)-Nets in Base 2

(237−89, 237, 77)-Net over F₂ — Constructive and digital

Digital (148, 237, 77)-net over F₂, using

3 times m-reduction [i] based on digital (148, 240, 77)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (48, 94, 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]
  2. digital (54, 146, 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]

(237−89, 237, 84)-Net in Base 2 — Constructive

(148, 237, 84)-net in base 2, using

5 times m-reduction [i] based on (148, 242, 84)-net in base 2, using
- trace code for nets [i] based on (27, 121, 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]

(237−89, 237, 123)-Net over F₂ — Digital

Digital (148, 237, 123)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(237−89, 237, 647)-Net in Base 2 — Upper bound on s

There is no (148, 237, 648)-net in base 2, because

1 times m-reduction [i] would yield (148, 236, 648)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 113276 258720 827476 007225 595549 257874 880906 639084 422840 444255 372206 266928 > 2²³⁶ [i]