MinT - Best Known (164−60, 164, s)-Nets in Base 4

Best Known (164−60, 164, s)-Nets in Base 4

(164−60, 164, 140)-Net over F₄ — Constructive and digital

Digital (104, 164, 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 (72, 132, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 66, 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]

(164−60, 164, 152)-Net in Base 4 — Constructive

(104, 164, 152)-net in base 4, using

trace code for nets [i] based on (22, 82, 76)-net in base 16, using
- 3 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]

(164−60, 164, 334)-Net over F₄ — Digital

Digital (104, 164, 334)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−60, 164, 7826)-Net in Base 4 — Upper bound on s

There is no (104, 164, 7827)-net in base 4, because

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]