MinT - Best Known (158−41, 158, s)-Nets in Base 4

Best Known (158−41, 158, s)-Nets in Base 4

(158−41, 158, 531)-Net over F₄ — Constructive and digital

Digital (117, 158, 531)-net over F₄, using

7 times m-reduction [i] based on digital (117, 165, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 55, 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]

(158−41, 158, 576)-Net in Base 4 — Constructive

(117, 158, 576)-net in base 4, using

1 times m-reduction [i] based on (117, 159, 576)-net in base 4, using
- trace code for nets [i] based on (11, 53, 192)-net in base 64, using
  - 3 times m-reduction [i] based on (11, 56, 192)-net in base 64, using
    - base change [i] based on digital (3, 48, 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]

(158−41, 158, 1275)-Net over F₄ — Digital

Digital (117, 158, 1275)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−41, 158, 147335)-Net in Base 4 — Upper bound on s

There is no (117, 158, 147336)-net in base 4, because

1 times m-reduction [i] would yield (117, 157, 147336)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 33375 744266 939419 677189 062123 732142 757979 459921 977760 323425 071304 965313 100481 727314 112566 299576 > 4¹⁵⁷ [i]