MinT - Best Known (240−99, 240, s)-Nets in Base 4

Best Known (240−99, 240, s)-Nets in Base 4

(240−99, 240, 137)-Net over F₄ — Constructive and digital

Digital (141, 240, 137)-net over F₄, using

7 times m-reduction [i] based on digital (141, 247, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 33)-net over F₄, using
    - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
        an explicitly constructive algebraic function field [i]
  2. digital (73, 179, 104)-net over F₄, using
    - net from sequence [i] based on digital (73, 103)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
        F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(240−99, 240, 311)-Net over F₄ — Digital

Digital (141, 240, 311)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(240−99, 240, 5464)-Net in Base 4 — Upper bound on s

There is no (141, 240, 5465)-net in base 4, because

1 times m-reduction [i] would yield (141, 239, 5465)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 781265 225877 563070 644992 353979 401169 873283 466754 657625 149339 993125 822506 000979 433192 914084 825179 552077 724005 328381 453529 602269 704467 464825 603528 > 4²³⁹ [i]