MinT - Best Known (156, 156+89, s)-Nets in Base 4

Best Known (156, 156+89, s)-Nets in Base 4

(156, 156+89, 160)-Net over F₄ — Constructive and digital

Digital (156, 245, 160)-net over F₄, using

11 times m-reduction [i] based on digital (156, 256, 160)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (33, 83, 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]
  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]

(156, 156+89, 196)-Net in Base 4 — Constructive

(156, 245, 196)-net in base 4, using

3 times m-reduction [i] based on (156, 248, 196)-net in base 4, using
- trace code for nets [i] based on (32, 124, 98)-net in base 16, using
  - 1 times m-reduction [i] based on (32, 125, 98)-net in base 16, using
    - base change [i] based on digital (7, 100, 98)-net over F₃₂, using
      - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(156, 156+89, 485)-Net over F₄ — Digital

Digital (156, 245, 485)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 156+89, 12509)-Net in Base 4 — Upper bound on s

There is no (156, 245, 12510)-net in base 4, because

1 times m-reduction [i] would yield (156, 244, 12510)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 800 131625 679919 591948 162146 862913 029055 867189 380104 007162 193419 179027 730344 974997 438842 795915 699571 114484 655520 040199 920659 935221 746151 270792 621282 > 4²⁴⁴ [i]