MinT - Best Known (219−59, 219, s)-Nets in Base 4

Best Known (219−59, 219, s)-Nets in Base 4

(219−59, 219, 531)-Net over F₄ — Constructive and digital

Digital (160, 219, 531)-net over F₄, using

t-expansion [i] based on digital (159, 219, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (159, 228, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 76, 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]

(219−59, 219, 576)-Net in Base 4 — Constructive

(160, 219, 576)-net in base 4, using

trace code for nets [i] based on (14, 73, 192)-net in base 64, using
- 4 times m-reduction [i] based on (14, 77, 192)-net in base 64, using
  - base change [i] based on digital (3, 66, 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]

(219−59, 219, 1433)-Net over F₄ — Digital

Digital (160, 219, 1433)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−59, 219, 130542)-Net in Base 4 — Upper bound on s

There is no (160, 219, 130543)-net in base 4, because

1 times m-reduction [i] would yield (160, 218, 130543)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177465 769087 035444 384561 377401 427340 825420 056913 370682 363222 445853 535546 269255 507054 402735 261452 212156 226696 408702 261320 127354 103972 > 4²¹⁸ [i]