MinT - Best Known (258−59, 258, s)-Nets in Base 2

Best Known (258−59, 258, s)-Nets in Base 2

(258−59, 258, 207)-Net over F₂ — Constructive and digital

Digital (199, 258, 207)-net over F₂, using

2¹ times duplication [i] based on digital (198, 257, 207)-net over F₂, using
- (u,Â u+v)-construction [i] based on
  1. digital (9, 38, 12)-net over F₂, using
    - net from sequence [i] based on digital (9, 11)-sequence over F₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂ with g(F) = 9 and N(F) ≥ 12, using
        an explicitly constructive algebraic function field [i]
  2. digital (160, 219, 195)-net over F₂, using
    - trace code for nets [i] based on digital (14, 73, 65)-net over F₈, using
      - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]

(258−59, 258, 440)-Net over F₂ — Digital

Digital (199, 258, 440)-net over F₂, using

Gilbertâ€“VarÅ¡amov bound [i]

(258−59, 258, 5388)-Net in Base 2 — Upper bound on s

There is no (199, 258, 5389)-net in base 2, because

1 times m-reduction [i] would yield (199, 257, 5389)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 232773 972839 462367 187304 533413 534241 785673 238171 388636 099891 974406 142667 585184 > 2²⁵⁷ [i]