MinT - Best Known (251−65, 251, s)-Nets in Base 4

Best Known (251−65, 251, s)-Nets in Base 4

(251−65, 251, 545)-Net over F₄ — Constructive and digital

Digital (186, 251, 545)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 35, 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 (151, 216, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 72, 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]

(251−65, 251, 648)-Net in Base 4 — Constructive

(186, 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]

(251−65, 251, 1921)-Net over F₄ — Digital

Digital (186, 251, 1921)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(251−65, 251, 215426)-Net in Base 4 — Upper bound on s

There is no (186, 251, 215427)-net in base 4, because

1 times m-reduction [i] would yield (186, 250, 215427)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 3 273410 992399 121685 229904 753357 218736 179721 292959 617864 443347 137056 380291 906102 389114 359380 257203 887847 099263 932890 480850 914971 248676 172926 720263 518100 > 4²⁵⁰ [i]