MinT - Best Known (151−39, 151, s)-Nets in Base 4

Best Known (151−39, 151, s)-Nets in Base 4

(151−39, 151, 531)-Net over F₄ — Constructive and digital

Digital (112, 151, 531)-net over F₄, using

t-expansion [i] based on digital (111, 151, 531)-net over F₄, using
- 5 times m-reduction [i] based on digital (111, 156, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 52, 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]

(151−39, 151, 576)-Net in Base 4 — Constructive

(112, 151, 576)-net in base 4, using

4⁴ times duplication [i] based on (108, 147, 576)-net in base 4, using
- trace code for nets [i] based on (10, 49, 192)-net in base 64, using
  - base change [i] based on digital (3, 42, 192)-net over F₁₂₈, using
    - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
        a function field by SÃ©mirat [i]

(151−39, 151, 1255)-Net over F₄ — Digital

Digital (112, 151, 1255)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(151−39, 151, 149676)-Net in Base 4 — Upper bound on s

There is no (112, 151, 149677)-net in base 4, because

1 times m-reduction [i] would yield (112, 150, 149677)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 037045 770341 087481 049692 860205 031738 890770 342189 632656 133595 922530 265389 730447 236831 414456 > 4¹⁵⁰ [i]