MinT - Best Known (234−96, 234, s)-Nets in Base 4

Best Known (234−96, 234, s)-Nets in Base 4

(234−96, 234, 137)-Net over F₄ — Constructive and digital

Digital (138, 234, 137)-net over F₄, using

4 times m-reduction [i] based on digital (138, 238, 137)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 65, 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, 173, 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]

(234−96, 234, 310)-Net over F₄ — Digital

Digital (138, 234, 310)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−96, 234, 5339)-Net in Base 4 — Upper bound on s

There is no (138, 234, 5340)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 764 002254 965260 523694 642042 872183 898810 498956 878951 370586 591281 184655 452224 683374 276611 427276 481655 565932 492683 852649 139078 442149 757590 249298 > 4²³⁴ [i]