MinT - Best Known (249−68, 249, s)-Nets in Base 4

Best Known (249−68, 249, s)-Nets in Base 4

(249−68, 249, 531)-Net over F₄ — Constructive and digital

Digital (181, 249, 531)-net over F₄, using

t-expansion [i] based on digital (179, 249, 531)-net over F₄, using
- 9 times m-reduction [i] based on digital (179, 258, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 86, 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]

(249−68, 249, 576)-Net in Base 4 — Constructive

(181, 249, 576)-net in base 4, using

trace code for nets [i] based on (15, 83, 192)-net in base 64, using
- 1 times m-reduction [i] based on (15, 84, 192)-net in base 64, using
  - base change [i] based on digital (3, 72, 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]

(249−68, 249, 1501)-Net over F₄ — Digital

Digital (181, 249, 1501)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(249−68, 249, 115732)-Net in Base 4 — Upper bound on s

There is no (181, 249, 115733)-net in base 4, because

the generalized Rao bound for nets shows that 4^mÂ â‰¥ 818565 886777 776156 913841 233178 195236 504949 168273 967814 996263 120252 600555 613166 183984 364487 012945 365265 900032 294597 936466 443378 666772 316462 605685 947072 > 4²⁴⁹ [i]