MinT - Best Known (56−27, 56, s)-Nets in Base 4

Best Known (56−27, 56, s)-Nets in Base 4

(56−27, 56, 48)-Net over F₄ — Constructive and digital

Digital (29, 56, 48)-net over F₄, using

trace code for nets [i] based on digital (1, 28, 24)-net over F₁₆, using
- net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
    - a function field by SÃ©mirat [i]

(56−27, 56, 59)-Net over F₄ — Digital

Digital (29, 56, 59)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(56−27, 56, 655)-Net in Base 4 — Upper bound on s

There is no (29, 56, 656)-net in base 4, because

1 times m-reduction [i] would yield (29, 55, 656)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1305 449564 455060 633586 184232 477255 > 4⁵⁵ [i]