MinT - Best Known (216−82, 216, s)-Nets in Base 4

Best Known (216−82, 216, s)-Nets in Base 4

(216−82, 216, 137)-Net over F₄ — Constructive and digital

Digital (134, 216, 137)-net over F₄, using

10 times m-reduction [i] based on digital (134, 226, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 61, 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, 165, 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]

(216−82, 216, 373)-Net over F₄ — Digital

Digital (134, 216, 373)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−82, 216, 7957)-Net in Base 4 — Upper bound on s

There is no (134, 216, 7958)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11105 892733 582657 914063 298245 762492 607724 518067 073625 546478 587646 174766 383472 479854 219911 117560 048875 364456 255852 377729 490652 967600 > 4²¹⁶ [i]