MinT - Best Known (105, 166, s)-Nets in Base 4

Best Known (105, 166, s)-Nets in Base 4

(105, 166, 140)-Net over F₄ — Constructive and digital

Digital (105, 166, 140)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (2, 32, 10)-net over F₄, using
  - net from sequence [i] based on digital (2, 9)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 2 and N(F) ≥ 10, using
      - an explicitly constructive algebraic function field [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]

(105, 166, 152)-Net in Base 4 — Constructive

(105, 166, 152)-net in base 4, using

trace code for nets [i] based on (22, 83, 76)-net in base 16, using
- 2 times m-reduction [i] based on (22, 85, 76)-net in base 16, using
  - base change [i] based on digital (5, 68, 76)-net over F₃₂, using
    - net from sequence [i] based on digital (5, 75)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 5 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(105, 166, 332)-Net over F₄ — Digital

Digital (105, 166, 332)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(105, 166, 8197)-Net in Base 4 — Upper bound on s

There is no (105, 166, 8198)-net in base 4, because

1 times m-reduction [i] would yield (105, 165, 8198)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2188 011630 392847 225144 532585 002819 185067 677891 769689 842615 581133 004999 290408 314541 796231 435513 589760 > 4¹⁶⁵ [i]