MinT - Best Known (235−85, 235, s)-Nets in Base 2

Best Known (235−85, 235, s)-Nets in Base 2

(235−85, 235, 84)-Net over F₂ — Constructive and digital

Digital (150, 235, 84)-net over F₂, using

(u,Â u+v)-construction [i] based on
1. digital (54, 96, 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, 139, 42)-net over F₂, using
  - net from sequence [i] based on digital (54, 41)-sequence over F₂ (see above)

(235−85, 235, 86)-Net in Base 2 — Constructive

(150, 235, 86)-net in base 2, using

5 times m-reduction [i] based on (150, 240, 86)-net in base 2, using
- trace code for nets [i] based on (30, 120, 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]

(235−85, 235, 133)-Net over F₂ — Digital

Digital (150, 235, 133)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(235−85, 235, 724)-Net in Base 2 — Upper bound on s

There is no (150, 235, 725)-net in base 2, because

1 times m-reduction [i] would yield (150, 234, 725)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 27810 794602 777597 632362 010950 955374 724644 217769 228082 381006 088889 211480 > 2²³⁴ [i]