MinT - Best Known (137, 137+47, s)-Nets in Base 4

Best Known (137, 137+47, s)-Nets in Base 4

(137, 137+47, 531)-Net over F₄ — Constructive and digital

Digital (137, 184, 531)-net over F₄, using

11 times m-reduction [i] based on digital (137, 195, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 65, 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]

(137, 137+47, 648)-Net in Base 4 — Constructive

(137, 184, 648)-net in base 4, using

4¹ times duplication [i] based on (136, 183, 648)-net in base 4, using
- trace code for nets [i] based on (14, 61, 216)-net in base 64, using
  - 2 times m-reduction [i] based on (14, 63, 216)-net in base 64, using
    - base change [i] based on digital (5, 54, 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]

(137, 137+47, 1559)-Net over F₄ — Digital

Digital (137, 184, 1559)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(137, 137+47, 193908)-Net in Base 4 — Upper bound on s

There is no (137, 184, 193909)-net in base 4, because

1 times m-reduction [i] would yield (137, 183, 193909)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 150 317364 233929 197347 059417 898064 713735 001505 145087 913003 393697 829336 324864 099805 909020 436408 550525 599362 136928 > 4¹⁸³ [i]