MinT - Best Known (189−65, 189, s)-Nets in Base 4

Best Known (189−65, 189, s)-Nets in Base 4

(189−65, 189, 163)-Net over F₄ — Constructive and digital

Digital (124, 189, 163)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (15, 47, 33)-net over F₄, using
  - net from sequence [i] based on digital (15, 32)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
      - an explicitly constructive algebraic function field [i]
2. digital (77, 142, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 71, 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]

(189−65, 189, 208)-Net in Base 4 — Constructive

(124, 189, 208)-net in base 4, using

t-expansion [i] based on (123, 189, 208)-net in base 4, using
- 1 times m-reduction [i] based on (123, 190, 208)-net in base 4, using
  - trace code for nets [i] based on (28, 95, 104)-net in base 16, using
    - base change [i] based on digital (9, 76, 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]

(189−65, 189, 467)-Net over F₄ — Digital

Digital (124, 189, 467)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(189−65, 189, 14658)-Net in Base 4 — Upper bound on s

There is no (124, 189, 14659)-net in base 4, because

1 times m-reduction [i] would yield (124, 188, 14659)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 154104 606796 407888 510127 734450 836076 214126 620623 431678 722437 213982 882605 500134 738578 665170 927684 748810 936848 687070 > 4¹⁸⁸ [i]