MinT - Best Known (186, 186+63, s)-Nets in Base 4

Best Known (186, 186+63, s)-Nets in Base 4

(186, 186+63, 552)-Net over F₄ — Constructive and digital

Digital (186, 249, 552)-net over F₄, using

4¹ times duplication [i] based on digital (185, 248, 552)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (7, 38, 21)-net over F₄, using
    - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, 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]

(186, 186+63, 648)-Net in Base 4 — Constructive

(186, 249, 648)-net in base 4, using

t-expansion [i] based on (185, 249, 648)-net in base 4, using
- 3 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]

(186, 186+63, 2119)-Net over F₄ — Digital

Digital (186, 249, 2119)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(186, 186+63, 271238)-Net in Base 4 — Upper bound on s

There is no (186, 249, 271239)-net in base 4, because

1 times m-reduction [i] would yield (186, 248, 271239)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 204602 356748 321465 562342 740422 631321 014189 834281 669473 242991 363207 399485 706155 904027 872721 588062 440958 352180 726063 271814 935447 369795 827598 487350 809920 > 4²⁴⁸ [i]