MinT - Best Known (252−101, 252, s)-Nets in Base 4

Best Known (252−101, 252, s)-Nets in Base 4

(252−101, 252, 138)-Net over F₄ — Constructive and digital

Digital (151, 252, 138)-net over F₄, using

t-expansion [i] based on digital (149, 252, 138)-net over F₄, using
- 7 times m-reduction [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]

(252−101, 252, 146)-Net in Base 4 — Constructive

(151, 252, 146)-net in base 4, using

1 times m-reduction [i] based on (151, 253, 146)-net in base 4, using
- (u,Â u+v)-construction [i] based on
  1. (27, 78, 42)-net in base 4, using
    - net from sequence [i] based on (27, 41)-sequence in base 4, using
      - base expansion [i] based on digital (54, 41)-sequence over F₂, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [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, 175, 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]

(252−101, 252, 357)-Net over F₄ — Digital

Digital (151, 252, 357)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(252−101, 252, 6796)-Net in Base 4 — Upper bound on s

There is no (151, 252, 6797)-net in base 4, because

1 times m-reduction [i] would yield (151, 251, 6797)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 13 172607 044019 594138 522361 975440 300006 645513 662007 581877 413057 047763 455356 756286 550072 148002 494567 367241 280540 860837 159558 164738 410895 861906 308076 344240 > 4²⁵¹ [i]