MinT - Best Known (161−41, 161, s)-Nets in Base 4

Best Known (161−41, 161, s)-Nets in Base 4

(161−41, 161, 531)-Net over F₄ — Constructive and digital

Digital (120, 161, 531)-net over F₄, using

t-expansion [i] based on digital (119, 161, 531)-net over F₄, using
- 7 times m-reduction [i] based on digital (119, 168, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 56, 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]

(161−41, 161, 576)-Net in Base 4 — Constructive

(120, 161, 576)-net in base 4, using

t-expansion [i] based on (119, 161, 576)-net in base 4, using
- 1 times m-reduction [i] based on (119, 162, 576)-net in base 4, using
  - trace code for nets [i] based on (11, 54, 192)-net in base 64, using
    - 2 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]

(161−41, 161, 1413)-Net over F₄ — Digital

Digital (120, 161, 1413)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(161−41, 161, 181395)-Net in Base 4 — Upper bound on s

There is no (120, 161, 181396)-net in base 4, because

1 times m-reduction [i] would yield (120, 160, 181396)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 136134 806232 593206 791682 132523 557451 821101 786324 905807 796353 086866 490012 563664 754699 518198 650416 > 4¹⁶⁰ [i]