MinT - Best Known (45, 45+31, s)-Nets in Base 4

Best Known (45, 45+31, s)-Nets in Base 4

(45, 45+31, 130)-Net over F₄ — Constructive and digital

Digital (45, 76, 130)-net over F₄, using

2 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]

(45, 45+31, 2180)-Net in Base 4 — Upper bound on s

There is no (45, 76, 2181)-net in base 4, because

1 times m-reduction [i] would yield (45, 75, 2181)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1431 604435 661986 620064 978936 463053 864332 066688 > 4⁷⁵ [i]