MinT - Best Known (245−93, 245, s)-Nets in Base 2

Best Known (245−93, 245, s)-Nets in Base 2

(245−93, 245, 78)-Net over F₂ — Constructive and digital

Digital (152, 245, 78)-net over F₂, using

1 times m-reduction [i] based on digital (152, 246, 78)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (51, 98, 36)-net over F₂, using
    - net from sequence [i] based on digital (51, 35)-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 3 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, 148, 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]

(245−93, 245, 84)-Net in Base 2 — Constructive

(152, 245, 84)-net in base 2, using

5 times m-reduction [i] based on (152, 250, 84)-net in base 2, using
- trace code for nets [i] based on (27, 125, 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]

(245−93, 245, 124)-Net over F₂ — Digital

Digital (152, 245, 124)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(245−93, 245, 645)-Net in Base 2 — Upper bound on s

There is no (152, 245, 646)-net in base 2, because

1 times m-reduction [i] would yield (152, 244, 646)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 29 050927 057531 996418 784352 333324 054835 645039 164370 010307 711572 303181 498304 > 2²⁴⁴ [i]