MinT - Best Known (205−75, 205, s)-Nets in Base 4

Best Known (205−75, 205, s)-Nets in Base 4

(205−75, 205, 147)-Net over F₄ — Constructive and digital

Digital (130, 205, 147)-net over F₄, using

4¹ times duplication [i] based on digital (129, 204, 147)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (5, 42, 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 (87, 162, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 81, 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]

(205−75, 205, 152)-Net in Base 4 — Constructive

(130, 205, 152)-net in base 4, using

3 times m-reduction [i] based on (130, 208, 152)-net in base 4, using
- trace code for nets [i] based on (26, 104, 76)-net in base 16, using
  - 1 times m-reduction [i] based on (26, 105, 76)-net in base 16, using
    - base change [i] based on digital (5, 84, 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]

(205−75, 205, 405)-Net over F₄ — Digital

Digital (130, 205, 405)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−75, 205, 10161)-Net in Base 4 — Upper bound on s

There is no (130, 205, 10162)-net in base 4, because

1 times m-reduction [i] would yield (130, 204, 10162)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 940450 564819 762588 002574 906570 133133 412087 225036 162337 018187 722552 646040 238316 413182 532530 829398 755145 783052 751287 568060 > 4²⁰⁴ [i]