MinT - Best Known (194−51, 194, s)-Nets in Base 4

Best Known (194−51, 194, s)-Nets in Base 4

(194−51, 194, 531)-Net over F₄ — Constructive and digital

Digital (143, 194, 531)-net over F₄, using

10 times m-reduction [i] based on digital (143, 204, 531)-net over F₄, using
- trace code for nets [i] based on digital (7, 68, 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]

(194−51, 194, 576)-Net in Base 4 — Constructive

(143, 194, 576)-net in base 4, using

1 times m-reduction [i] based on (143, 195, 576)-net in base 4, using
- trace code for nets [i] based on (13, 65, 192)-net in base 64, using
  - 5 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
    - base change [i] based on digital (3, 60, 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]

(194−51, 194, 1433)-Net over F₄ — Digital

Digital (143, 194, 1433)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(194−51, 194, 150781)-Net in Base 4 — Upper bound on s

There is no (143, 194, 150782)-net in base 4, because

1 times m-reduction [i] would yield (143, 193, 150782)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 157 611225 705076 818852 732598 857810 879050 434050 336519 652266 801716 438187 314797 393366 388815 436586 245833 457589 390543 732487 > 4¹⁹³ [i]