MinT - Best Known (230−90, 230, s)-Nets in Base 4

Best Known (230−90, 230, s)-Nets in Base 4

(230−90, 230, 138)-Net over F₄ — Constructive and digital

Digital (140, 230, 138)-net over F₄, using

2 times m-reduction [i] based on digital (140, 232, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 67, 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, 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]

(230−90, 230, 356)-Net over F₄ — Digital

Digital (140, 230, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(230−90, 230, 6981)-Net in Base 4 — Upper bound on s

There is no (140, 230, 6982)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 984166 853463 748268 176563 292730 958692 165214 555654 765990 463339 719057 691739 861745 014470 200563 348751 166222 392488 187372 460246 182249 563890 601384 > 4²³⁰ [i]