MinT - Best Known (214−57, 214, s)-Nets in Base 4

Best Known (214−57, 214, s)-Nets in Base 4

(214−57, 214, 531)-Net over F₄ — Constructive and digital

Digital (157, 214, 531)-net over F₄, using

11 times m-reduction [i] based on digital (157, 225, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 75, 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]

(214−57, 214, 576)-Net in Base 4 — Constructive

(157, 214, 576)-net in base 4, using

4⁴ times duplication [i] based on (153, 210, 576)-net in base 4, using
- trace code for nets [i] based on (13, 70, 192)-net in base 64, using
  - base change [i] based on digital (3, 60, 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]

(214−57, 214, 1474)-Net over F₄ — Digital

Digital (157, 214, 1474)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(214−57, 214, 143143)-Net in Base 4 — Upper bound on s

There is no (157, 214, 143144)-net in base 4, because

1 times m-reduction [i] would yield (157, 213, 143144)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 173 317303 518062 535650 776624 433615 259781 914545 411028 155070 949039 084289 324375 393495 588534 792556 699664 449955 821677 210516 174355 610828 > 4²¹³ [i]