MinT - Best Known (165−65, 165, s)-Nets in Base 4

Best Known (165−65, 165, s)-Nets in Base 4

(165−65, 165, 130)-Net over F₄ — Constructive and digital

Digital (100, 165, 130)-net over F₄, using

23 times m-reduction [i] based on digital (100, 188, 130)-net over F₄, using
- trace code for nets [i] based on digital (6, 94, 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]

(165−65, 165, 260)-Net over F₄ — Digital

Digital (100, 165, 260)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−65, 165, 5165)-Net in Base 4 — Upper bound on s

There is no (100, 165, 5166)-net in base 4, because

1 times m-reduction [i] would yield (100, 164, 5166)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 547 935124 266588 968303 863023 179656 087079 015997 616049 360441 052670 300618 948714 413452 208395 508279 340021 > 4¹⁶⁴ [i]