MinT - Best Known (254−100, 254, s)-Nets in Base 2

Best Known (254−100, 254, s)-Nets in Base 2

(254−100, 254, 77)-Net over F₂ — Constructive and digital

Digital (154, 254, 77)-net over F₂, using

4 times m-reduction [i] based on digital (154, 258, 77)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (48, 100, 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, 158, 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]

(254−100, 254, 84)-Net in Base 2 — Constructive

(154, 254, 84)-net in base 2, using

trace code for nets [i] based on (27, 127, 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]

(254−100, 254, 117)-Net over F₂ — Digital

Digital (154, 254, 117)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−100, 254, 588)-Net in Base 2 — Upper bound on s

There is no (154, 254, 589)-net in base 2, because

the generalized Rao bound for nets shows that 2^mÂ â‰¥ 30895 885082 260820 062097 213240 168731 761475 342235 793714 538815 644281 910230 407384 > 2²⁵⁴ [i]