MinT - Best Known (216−77, 216, s)-Nets in Base 4

Best Known (216−77, 216, s)-Nets in Base 4

(216−77, 216, 158)-Net over F₄ — Constructive and digital

Digital (139, 216, 158)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (12, 50, 28)-net over F₄, using
  - net from sequence [i] based on digital (12, 27)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 12 and N(F) ≥ 28, using
      - an explicitly constructive algebraic function field [i]
2. digital (89, 166, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 83, 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]

(216−77, 216, 208)-Net in Base 4 — Constructive

(139, 216, 208)-net in base 4, using

trace code for nets [i] based on (31, 108, 104)-net in base 16, using
- 2 times m-reduction [i] based on (31, 110, 104)-net in base 16, using
  - base change [i] based on digital (9, 88, 104)-net over F₃₂, using
    - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
        a function field by SÃ©mirat [i]

(216−77, 216, 464)-Net over F₄ — Digital

Digital (139, 216, 464)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(216−77, 216, 12735)-Net in Base 4 — Upper bound on s

There is no (139, 216, 12736)-net in base 4, because

1 times m-reduction [i] would yield (139, 215, 12736)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2777 420236 340078 075314 842087 036890 434965 736830 773620 945556 451960 813483 656022 663864 760443 370665 532311 133392 400899 079231 934191 582447 > 4²¹⁵ [i]