MinT - Best Known (158−40, 158, s)-Nets in Base 4

Best Known (158−40, 158, s)-Nets in Base 4

(158−40, 158, 531)-Net over F₄ — Constructive and digital

Digital (118, 158, 531)-net over F₄, using

t-expansion [i] based on 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]

(158−40, 158, 576)-Net in Base 4 — Constructive

(118, 158, 576)-net in base 4, using

t-expansion [i] based on (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]

(158−40, 158, 1430)-Net over F₄ — Digital

Digital (118, 158, 1430)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(158−40, 158, 157911)-Net in Base 4 — Upper bound on s

There is no (118, 158, 157912)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 133503 397730 936641 059819 159706 462595 926046 122121 663947 360047 256502 162410 305537 328694 188874 290471 > 4¹⁵⁸ [i]