MinT - Best Known (219−55, 219, s)-Nets in Base 4

Best Known (219−55, 219, s)-Nets in Base 4

(219−55, 219, 548)-Net over F₄ — Constructive and digital

Digital (164, 219, 548)-net over F₄, using

4¹ times duplication [i] based on digital (163, 218, 548)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 32, 17)-net over F₄, using
    - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
        an explicitly constructive algebraic function field [i]
  2. digital (131, 186, 531)-net over F₄, using
    - trace code for nets [i] based on digital (7, 62, 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]

(219−55, 219, 648)-Net in Base 4 — Constructive

(164, 219, 648)-net in base 4, using

3 times m-reduction [i] based on (164, 222, 648)-net in base 4, using
- trace code for nets [i] based on (16, 74, 216)-net in base 64, using
  - 3 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
    - base change [i] based on digital (5, 66, 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]

(219−55, 219, 1959)-Net over F₄ — Digital

Digital (164, 219, 1959)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−55, 219, 264439)-Net in Base 4 — Upper bound on s

There is no (164, 219, 264440)-net in base 4, because

1 times m-reduction [i] would yield (164, 218, 264440)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 177459 625445 008610 912286 076692 764956 822960 488411 743934 386487 352934 007193 401467 710603 574197 021382 433770 431980 011225 868335 579910 939820 > 4²¹⁸ [i]