MinT - Best Known (250−100, 250, s)-Nets in Base 4

Best Known (250−100, 250, s)-Nets in Base 4

(250−100, 250, 138)-Net over F₄ — Constructive and digital

Digital (150, 250, 138)-net over F₄, using

t-expansion [i] based on digital (149, 250, 138)-net over F₄, using
- 9 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]

(250−100, 250, 146)-Net in Base 4 — Constructive

(150, 250, 146)-net in base 4, using

(u,Â u+v)-construction [i] based on
1. (27, 77, 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, 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]

(250−100, 250, 356)-Net over F₄ — Digital

Digital (150, 250, 356)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(250−100, 250, 6609)-Net in Base 4 — Upper bound on s

There is no (150, 250, 6610)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 292775 304388 165923 246599 864988 768733 538749 474330 762108 466073 655846 156971 526164 048204 691039 118526 335576 296758 243537 145844 257807 083264 496685 725601 235856 > 4²⁵⁰ [i]