MinT - Best Known (216−55, 216, s)-Nets in Base 4

Best Known (216−55, 216, s)-Nets in Base 4

(216−55, 216, 545)-Net over F₄ — Constructive and digital

Digital (161, 216, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 30, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (131, 186, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 62, 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]

(216−55, 216, 648)-Net in Base 4 — Constructive

(161, 216, 648)-net in base 4, using

t-expansion [i] based on (160, 216, 648)-net in base 4, using
- trace code for nets [i] based on (16, 72, 216)-net in base 64, using
  - 5 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 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]

(216−55, 216, 1816)-Net over F₄ — Digital

Digital (161, 216, 1816)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−55, 216, 226686)-Net in Base 4 — Upper bound on s

There is no (161, 216, 226687)-net in base 4, because

1 times m-reduction [i] would yield (161, 215, 226687)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2772 985501 637620 301060 797221 825457 047846 454399 833123 853182 118487 920834 082334 872422 481495 855551 355420 579778 884595 803603 883666 823600 > 4²¹⁵ [i]