MinT - Best Known (234−92, 234, s)-Nets in Base 4

Best Known (234−92, 234, s)-Nets in Base 4

(234−92, 234, 138)-Net over F₄ — Constructive and digital

Digital (142, 234, 138)-net over F₄, using

4 times m-reduction [i] based on digital (142, 238, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 69, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (73, 169, 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]

(234−92, 234, 355)-Net over F₄ — Digital

Digital (142, 234, 355)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(234−92, 234, 6892)-Net in Base 4 — Upper bound on s

There is no (142, 234, 6893)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 763 680472 609355 515447 416741 279533 307871 635461 117511 970007 924352 765508 926907 753671 541544 235870 698817 745933 482261 297856 523731 366729 602289 355440 > 4²³⁴ [i]