MinT - Best Known (57−25, 57, s)-Nets in Base 4

Best Known (57−25, 57, s)-Nets in Base 4

(57−25, 57, 76)-Net over F₄ — Constructive and digital

Digital (32, 57, 76)-net over F₄, using

1 times m-reduction [i] based on digital (32, 58, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 29, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(57−25, 57, 80)-Net over F₄ — Digital

Digital (32, 57, 80)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−25, 57, 1127)-Net in Base 4 — Upper bound on s

There is no (32, 57, 1128)-net in base 4, because

1 times m-reduction [i] would yield (32, 56, 1128)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5204 672147 438093 873033 016843 055348 > 4⁵⁶ [i]