MinT - Best Known (228−84, 228, s)-Nets in Base 4

Best Known (228−84, 228, s)-Nets in Base 4

(228−84, 228, 147)-Net over F₄ — Constructive and digital

Digital (144, 228, 147)-net over F₄, using

1 times m-reduction [i] based on 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]

(228−84, 228, 196)-Net in Base 4 — Constructive

(144, 228, 196)-net in base 4, using

trace code for nets [i] based on (30, 114, 98)-net in base 16, using
- 1 times m-reduction [i] based on (30, 115, 98)-net in base 16, using
  - base change [i] based on digital (7, 92, 98)-net over F₃₂, using
    - net from sequence [i] based on digital (7, 97)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 7 and N(F) ≥ 98, using
        a function field by SÃ©mirat [i]

(228−84, 228, 433)-Net over F₄ — Digital

Digital (144, 228, 433)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(228−84, 228, 10175)-Net in Base 4 — Upper bound on s

There is no (144, 228, 10176)-net in base 4, because

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]