MinT - Best Known (189−69, 189, s)-Nets in Base 4

Best Known (189−69, 189, s)-Nets in Base 4

(189−69, 189, 147)-Net over F₄ — Constructive and digital

Digital (120, 189, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 39, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (81, 150, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 75, 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]

(189−69, 189, 152)-Net in Base 4 — Constructive

(120, 189, 152)-net in base 4, using

t-expansion [i] based on (119, 189, 152)-net in base 4, using
- 1 times m-reduction [i] based on (119, 190, 152)-net in base 4, using
  - trace code for nets [i] based on (24, 95, 76)-net in base 16, using
    - base change [i] based on digital (5, 76, 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]

(189−69, 189, 381)-Net over F₄ — Digital

Digital (120, 189, 381)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−69, 189, 9596)-Net in Base 4 — Upper bound on s

There is no (120, 189, 9597)-net in base 4, because

1 times m-reduction [i] would yield (120, 188, 9597)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 153914 708178 644115 769181 199138 003009 510171 215053 104352 091242 155243 847019 977582 115825 227455 027724 256140 894830 608242 > 4¹⁸⁸ [i]