MinT - Best Known (72−27, 72, s)-Nets in Base 4

Best Known (72−27, 72, s)-Nets in Base 4

(72−27, 72, 130)-Net over F₄ — Constructive and digital

Digital (45, 72, 130)-net over F₄, using

6 times m-reduction [i] based on digital (45, 78, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 39, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(72−27, 72, 159)-Net over F₄ — Digital

Digital (45, 72, 159)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−27, 72, 3657)-Net in Base 4 — Upper bound on s

There is no (45, 72, 3658)-net in base 4, because

1 times m-reduction [i] would yield (45, 71, 3658)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 576642 905095 080913 662380 715415 795785 890480 > 4⁷¹ [i]