MinT - Best Known (229−85, 229, s)-Nets in Base 4

Best Known (229−85, 229, s)-Nets in Base 4

(229−85, 229, 147)-Net over F₄ — Constructive and digital

Digital (144, 229, 147)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 47, 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 (97, 182, 130)-net over F₄, using
  - trace code for nets [i] based on digital (6, 91, 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]

(229−85, 229, 152)-Net in Base 4 — Constructive

(144, 229, 152)-net in base 4, using

t-expansion [i] based on (143, 229, 152)-net in base 4, using
- 1 times m-reduction [i] based on (143, 230, 152)-net in base 4, using
  - trace code for nets [i] based on (28, 115, 76)-net in base 16, using
    - base change [i] based on digital (5, 92, 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]

(229−85, 229, 424)-Net over F₄ — Digital

Digital (144, 229, 424)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(229−85, 229, 10175)-Net in Base 4 — Upper bound on s

There is no (144, 229, 10176)-net in base 4, because

1 times m-reduction [i] would yield (144, 228, 10176)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 186770 907778 594687 820751 596843 628357 985979 264392 890796 741081 733783 721985 216118 663825 526182 670980 656373 122730 006107 519463 929773 491338 434775 > 4²²⁸ [i]