MinT - Best Known (241−151, 241, s)-Nets in Base 2

Best Known (241−151, 241, s)-Nets in Base 2

(241−151, 241, 53)-Net over F₂ — Constructive and digital

Digital (90, 241, 53)-net over F₂, using

net from sequence [i] based on digital (90, 52)-sequence over F₂, using
- Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 69, N(F)Â =Â 48, 1 place with degree 2, and 4 places with degree 6 [i] based on function field F/F₂ with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]

(241−151, 241, 57)-Net over F₂ — Digital

Digital (90, 241, 57)-net over F₂, using

t-expansion [i] based on digital (83, 241, 57)-net over F₂, using
- net from sequence [i] based on digital (83, 56)-sequence over F₂, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 83 and N(F) ≥ 57, using
    - an algebraic function field by Auer [i]

(241−151, 241, 169)-Net in Base 2 — Upper bound on s

There is no (90, 241, 170)-net in base 2, because

1 times m-reduction [i] would yield (90, 240, 170)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 1 787788 330360 166558 464304 090476 022105 458004 142529 301990 693068 549895 509000 > 2²⁴⁰ [i]