MinT - Best Known (189, 189+69, s)-Nets in Base 4

Best Known (189, 189+69, s)-Nets in Base 4

(189, 189+69, 531)-Net over F₄ — Constructive and digital

Digital (189, 258, 531)-net over F₄, using

t-expansion [i] based on digital (179, 258, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 86, 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]

(189, 189+69, 576)-Net in Base 4 — Constructive

(189, 258, 576)-net in base 4, using

t-expansion [i] based on (188, 258, 576)-net in base 4, using
- trace code for nets [i] based on (16, 86, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (16, 91, 192)-net in base 64, using
    - base change [i] based on digital (3, 78, 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]

(189, 189+69, 1702)-Net over F₄ — Digital

Digital (189, 258, 1702)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(189, 189+69, 160377)-Net in Base 4 — Upper bound on s

There is no (189, 258, 160378)-net in base 4, because

1 times m-reduction [i] would yield (189, 257, 160378)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 53634 378431 083062 156965 577188 724366 951863 147059 588241 592278 067143 535661 581463 760702 152090 755558 369880 888376 081979 403341 856863 280220 986951 172403 646617 339076 > 4²⁵⁷ [i]