MinT - Best Known (72, 72+67, s)-Nets in Base 4

Best Known (72, 72+67, s)-Nets in Base 4

(72, 72+67, 73)-Net over F₄ — Constructive and digital

Digital (72, 139, 73)-net over F₄, using

1 times m-reduction [i] based on digital (72, 140, 73)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 39, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (33, 101, 56)-net over F₄, using
    - net from sequence [i] based on digital (33, 55)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
        F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(72, 72+67, 118)-Net over F₄ — Digital

Digital (72, 139, 118)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(72, 72+67, 1418)-Net in Base 4 — Upper bound on s

There is no (72, 139, 1419)-net in base 4, because

1 times m-reduction [i] would yield (72, 138, 1419)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 123017 993613 844570 926949 918910 981990 744554 802729 865918 542361 529319 087687 765447 520712 > 4¹³⁸ [i]