MinT - Best Known (54, 54+37, s)-Nets in Base 4

Best Known (54, 54+37, s)-Nets in Base 4

(54, 54+37, 130)-Net over F₄ — Constructive and digital

Digital (54, 91, 130)-net over F₄, using

5 times m-reduction [i] based on digital (54, 96, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 48, 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]

(54, 54+37, 145)-Net over F₄ — Digital

Digital (54, 91, 145)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+37, 2563)-Net in Base 4 — Upper bound on s

There is no (54, 91, 2564)-net in base 4, because

1 times m-reduction [i] would yield (54, 90, 2564)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1 535870 165513 871727 238700 965881 838680 814548 448745 436960 > 4⁹⁰ [i]