MinT - Best Known (152, 152+99, s)-Nets in Base 4

Best Known (152, 152+99, s)-Nets in Base 4

(152, 152+99, 138)-Net over F₄ — Constructive and digital

Digital (152, 251, 138)-net over F₄, using

t-expansion [i] based on digital (149, 251, 138)-net over F₄, using
- 8 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]

(152, 152+99, 147)-Net in Base 4 — Constructive

(152, 251, 147)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (30, 79, 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, 172, 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]

(152, 152+99, 374)-Net over F₄ — Digital

Digital (152, 251, 374)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(152, 152+99, 7474)-Net in Base 4 — Upper bound on s

There is no (152, 251, 7475)-net in base 4, because

1 times m-reduction [i] would yield (152, 250, 7475)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 281457 872235 591204 672433 477726 420986 172288 691213 236523 547223 950148 127304 206122 100941 326878 883474 663589 431543 792809 717254 777620 649800 665446 970597 444938 > 4²⁵⁰ [i]