MinT - Best Known (168, 168+67, s)-Nets in Base 4

Best Known (168, 168+67, s)-Nets in Base 4

(168, 168+67, 531)-Net over F₄ — Constructive and digital

Digital (168, 235, 531)-net over F₄, using

t-expansion [i] based on digital (167, 235, 531)-net over F₄, using
- 5 times m-reduction [i] based on digital (167, 240, 531)-net over F₄, using
  - trace code for nets [i] based on digital (7, 80, 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]

(168, 168+67, 1180)-Net over F₄ — Digital

Digital (168, 235, 1180)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(168, 168+67, 81515)-Net in Base 4 — Upper bound on s

There is no (168, 235, 81516)-net in base 4, because

1 times m-reduction [i] would yield (168, 234, 81516)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 762 211908 312291 840222 815500 594739 041745 915606 707666 417317 303025 638181 042592 417115 040458 997532 454499 820286 873248 877055 905319 617606 338223 092143 > 4²³⁴ [i]