MinT - Best Known (255−95, 255, s)-Nets in Base 2

Best Known (255−95, 255, s)-Nets in Base 2

(255−95, 255, 85)-Net over F₂ — Constructive and digital

Digital (160, 255, 85)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (54, 101, 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 (59, 154, 43)-net over F₂, using
  - net from sequence [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]

(255−95, 255, 86)-Net in Base 2 — Constructive

(160, 255, 86)-net in base 2, using

5 times m-reduction [i] based on (160, 260, 86)-net in base 2, using
- trace code for nets [i] based on (30, 130, 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]

(255−95, 255, 133)-Net over F₂ — Digital

Digital (160, 255, 133)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(255−95, 255, 710)-Net in Base 2 — Upper bound on s

There is no (160, 255, 711)-net in base 2, because

1 times m-reduction [i] would yield (160, 254, 711)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 29509 125075 323033 591298 750240 066052 931375 335074 102759 128606 331884 307281 422960 > 2²⁵⁴ [i]