MinT - Best Known (188, 188+63, s)-Nets in Base 4

Best Known (188, 188+63, s)-Nets in Base 4

(188, 188+63, 558)-Net over F₄ — Constructive and digital

Digital (188, 251, 558)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 41, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (147, 210, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 70, 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]

(188, 188+63, 648)-Net in Base 4 — Constructive

(188, 251, 648)-net in base 4, using

t-expansion [i] based on (185, 251, 648)-net in base 4, using
- 1 times m-reduction [i] based on (185, 252, 648)-net in base 4, using
  - trace code for nets [i] based on (17, 84, 216)-net in base 64, using
    - base change [i] based on digital (5, 72, 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]

(188, 188+63, 2215)-Net over F₄ — Digital

Digital (188, 251, 2215)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(188, 188+63, 296617)-Net in Base 4 — Upper bound on s

There is no (188, 251, 296618)-net in base 4, because

1 times m-reduction [i] would yield (188, 250, 296618)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 273478 252927 104983 397265 949138 878616 256053 864077 646317 398806 654302 488816 429187 322289 679374 458289 231276 006503 464725 781850 590750 607233 272693 101054 188720 > 4²⁵⁰ [i]