MinT - Best Known (57, 57+31, s)-Nets in Base 4

Best Known (57, 57+31, s)-Nets in Base 4

(57, 57+31, 130)-Net over F₄ — Constructive and digital

Digital (57, 88, 130)-net over F₄, using

14 times m-reduction [i] based on digital (57, 102, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 51, 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]

(57, 57+31, 231)-Net over F₄ — Digital

Digital (57, 88, 231)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+31, 6634)-Net in Base 4 — Upper bound on s

There is no (57, 88, 6635)-net in base 4, because

1 times m-reduction [i] would yield (57, 87, 6635)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 23971 551407 960437 652177 996762 590283 727908 850885 022528 > 4⁸⁷ [i]