MinT - Best Known (165−75, 165, s)-Nets in Base 4

Best Known (165−75, 165, s)-Nets in Base 4

(165−75, 165, 130)-Net over F₄ — Constructive and digital

Digital (90, 165, 130)-net over F₄, using

3 times m-reduction [i] based on digital (90, 168, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 84, 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]

(165−75, 165, 164)-Net over F₄ — Digital

Digital (90, 165, 164)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−75, 165, 2246)-Net in Base 4 — Upper bound on s

There is no (90, 165, 2247)-net in base 4, because

1 times m-reduction [i] would yield (90, 164, 2247)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 547 580231 306067 368256 019225 446328 066005 628049 347142 871866 147501 062527 503299 137731 292243 501803 908060 > 4¹⁶⁴ [i]