MinT - Best Known (165−61, 165, s)-Nets in Base 4

Best Known (165−61, 165, s)-Nets in Base 4

(165−61, 165, 139)-Net over F₄ — Constructive and digital

Digital (104, 165, 139)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (1, 31, 9)-net over F₄, using
  - net from sequence [i] based on digital (1, 8)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 1 and N(F) ≥ 9, using
      - the Hermitian function field over F₄ [i]
2. digital (73, 134, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 67, 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]

(165−61, 165, 324)-Net over F₄ — Digital

Digital (104, 165, 324)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−61, 165, 7826)-Net in Base 4 — Upper bound on s

There is no (104, 165, 7827)-net in base 4, because

1 times m-reduction [i] would yield (104, 164, 7827)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 547 656088 491375 232892 409343 073580 196356 293549 329635 676694 495149 289064 593820 019279 756529 527957 487392 > 4¹⁶⁴ [i]