MinT - Best Known (205−71, 205, s)-Nets in Base 4

Best Known (205−71, 205, s)-Nets in Base 4

(205−71, 205, 163)-Net over F₄ — Constructive and digital

Digital (134, 205, 163)-net over F₄, using

4¹ times duplication [i] based on digital (133, 204, 163)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 50, 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 (83, 154, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 77, 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]

(205−71, 205, 208)-Net in Base 4 — Constructive

(134, 205, 208)-net in base 4, using

3 times m-reduction [i] based on (134, 208, 208)-net in base 4, using
- trace code for nets [i] based on (30, 104, 104)-net in base 16, using
  - 1 times m-reduction [i] based on (30, 105, 104)-net in base 16, using
    - base change [i] based on digital (9, 84, 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]

(205−71, 205, 490)-Net over F₄ — Digital

Digital (134, 205, 490)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(205−71, 205, 14943)-Net in Base 4 — Upper bound on s

There is no (134, 205, 14944)-net in base 4, because

1 times m-reduction [i] would yield (134, 204, 14944)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 661 394171 511160 635243 284094 335425 798954 887890 579704 134051 278080 944456 949443 497189 239977 258069 695617 754090 021182 417820 191930 > 4²⁰⁴ [i]