MinT - Best Known (233−87, 233, s)-Nets in Base 2

Best Known (233−87, 233, s)-Nets in Base 2

(233−87, 233, 77)-Net over F₂ — Constructive and digital

Digital (146, 233, 77)-net over F₂, using

1 times m-reduction [i] based on digital (146, 234, 77)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (48, 92, 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, 142, 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]

(233−87, 233, 84)-Net in Base 2 — Constructive

(146, 233, 84)-net in base 2, using

5 times m-reduction [i] based on (146, 238, 84)-net in base 2, using
- trace code for nets [i] based on (27, 119, 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]

(233−87, 233, 123)-Net over F₂ — Digital

Digital (146, 233, 123)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−87, 233, 649)-Net in Base 2 — Upper bound on s

There is no (146, 233, 650)-net in base 2, because

1 times m-reduction [i] would yield (146, 232, 650)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 7278 573335 161608 292357 445706 483221 898539 688175 924233 560424 423883 234992 > 2²³² [i]