MinT - Best Known (209−55, 209, s)-Nets in Base 4

Best Known (209−55, 209, s)-Nets in Base 4

(209−55, 209, 531)-Net over F₄ — Constructive and digital

Digital (154, 209, 531)-net over F₄, using

t-expansion [i] based on digital (153, 209, 531)-net over F₄, using
- 10 times m-reduction [i] based on digital (153, 219, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 73, 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]

(209−55, 209, 576)-Net in Base 4 — Constructive

(154, 209, 576)-net in base 4, using

t-expansion [i] based on (153, 209, 576)-net in base 4, using
- 1 times m-reduction [i] based on (153, 210, 576)-net in base 4, using
  - trace code for nets [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]

(209−55, 209, 1522)-Net over F₄ — Digital

Digital (154, 209, 1522)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(209−55, 209, 158240)-Net in Base 4 — Upper bound on s

There is no (154, 209, 158241)-net in base 4, because

1 times m-reduction [i] would yield (154, 208, 158241)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 169252 321225 578560 181545 295002 435822 167111 371875 644941 265753 231129 822238 651376 078046 963394 781908 214203 369688 859374 200170 427296 > 4²⁰⁸ [i]