MinT - Best Known (156, 156+103, s)-Nets in Base 4

Best Known (156, 156+103, s)-Nets in Base 4

(156, 156+103, 138)-Net over F₄ — Constructive and digital

Digital (156, 259, 138)-net over F₄, using

t-expansion [i] based on digital (149, 259, 138)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 76, 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, 183, 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+103, 147)-Net in Base 4 — Constructive

(156, 259, 147)-net in base 4, using

t-expansion [i] based on (155, 259, 147)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (30, 82, 43)-net in base 4, using
    - net from sequence [i] based on (30, 42)-sequence in base 4, using
      - base expansion [i] based on digital (60, 42)-sequence over F₂, using
        t-expansion [i] based on digital (59, 42)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₂ with g(F)Â =Â 54, N(F)Â =Â 42, and 1 place with degree 6 [i] based on function field F/F₂ with g(F) = 54 and N(F) ≥ 42, using an explicitly constructive algebraic function field [i]
  2. digital (73, 177, 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+103, 375)-Net over F₄ — Digital

Digital (156, 259, 375)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(156, 156+103, 7310)-Net in Base 4 — Upper bound on s

There is no (156, 259, 7311)-net in base 4, because

1 times m-reduction [i] would yield (156, 258, 7311)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 214761 354626 108311 027287 370847 745509 974231 853733 367634 176016 361838 382335 642529 553537 842903 411920 011348 553764 725914 360708 636327 692044 951973 322025 789783 712832 > 4²⁵⁸ [i]