MinT - Best Known (235−87, 235, s)-Nets in Base 2

Best Known (235−87, 235, s)-Nets in Base 2

(235−87, 235, 78)-Net over F₂ — Constructive and digital

Digital (148, 235, 78)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (51, 94, 36)-net over F₂, using
  - net from sequence [i] based on digital (51, 35)-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 3 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, 141, 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]

(235−87, 235, 86)-Net in Base 2 — Constructive

(148, 235, 86)-net in base 2, using

1 times m-reduction [i] based on (148, 236, 86)-net in base 2, using
- trace code for nets [i] based on (30, 118, 43)-net in base 4, using
  - net from sequence [i] based on (30, 42)-sequence in base 4, using
    - base expansion [i] based on digital (60, 42)-sequence over F₂, using
      - t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]

(235−87, 235, 126)-Net over F₂ — Digital

Digital (148, 235, 126)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−87, 235, 672)-Net in Base 2 — Upper bound on s

There is no (148, 235, 673)-net in base 2, because

1 times m-reduction [i] would yield (148, 234, 673)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 28714 033700 333978 183039 849590 317165 137343 919974 030489 708404 088905 966464 > 2²³⁴ [i]