MinT - Best Known (254−95, 254, s)-Nets in Base 2

Best Known (254−95, 254, s)-Nets in Base 2

(254−95, 254, 84)-Net over F₂ — Constructive and digital

Digital (159, 254, 84)-net over F₂, using

t-expansion [i] based on digital (158, 254, 84)-net over F₂, using
- 4 times m-reduction [i] based on digital (158, 258, 84)-net over F₂, using
  - (u,Â u+v)-construction [i] based on
    1. digital (54, 104, 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]
    2. digital (54, 154, 42)-net over F₂, using
      - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(254−95, 254, 86)-Net in Base 2 — Constructive

(159, 254, 86)-net in base 2, using

4 times m-reduction [i] based on (159, 258, 86)-net in base 2, using
- trace code for nets [i] based on (30, 129, 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]

(254−95, 254, 132)-Net over F₂ — Digital

Digital (159, 254, 132)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(254−95, 254, 699)-Net in Base 2 — Upper bound on s

There is no (159, 254, 700)-net in base 2, because

1 times m-reduction [i] would yield (159, 253, 700)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 15079 674638 099367 564149 107791 998620 014667 982256 578773 420745 952892 154052 710004 > 2²⁵³ [i]