MinT - Best Known (150, 150+51, s)-Nets in Base 4

Best Known (150, 150+51, s)-Nets in Base 4

(150, 150+51, 541)-Net over F₄ — Constructive and digital

Digital (150, 201, 541)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 27, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [i]
2. digital (123, 174, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 58, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(150, 150+51, 648)-Net in Base 4 — Constructive

(150, 201, 648)-net in base 4, using

t-expansion [i] based on (149, 201, 648)-net in base 4, using
- trace code for nets [i] based on (15, 67, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (15, 70, 216)-net in base 64, using
    - base change [i] based on digital (5, 60, 216)-net over F₁₂₈, using
      - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(150, 150+51, 1734)-Net over F₄ — Digital

Digital (150, 201, 1734)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(150, 150+51, 222302)-Net in Base 4 — Upper bound on s

There is no (150, 201, 222303)-net in base 4, because

1 times m-reduction [i] would yield (150, 200, 222303)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 582330 269505 148441 512510 715243 110465 906624 498847 063475 725311 266485 378799 282796 907844 506665 430038 843429 299719 781872 434530 > 4²⁰⁰ [i]