MinT - Best Known (226−77, 226, s)-Nets in Base 4

Best Known (226−77, 226, s)-Nets in Base 4

(226−77, 226, 164)-Net over F₄ — Constructive and digital

Digital (149, 226, 164)-net over F₄, using

4¹ times duplication [i] based on digital (148, 225, 164)-net over F₄, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 59, 34)-net over F₄, using
    - net from sequence [i] based on digital (21, 33)-sequence over F₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
        T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]
  2. digital (89, 166, 130)-net over F₄, using
    - trace code for nets [i] based on digital (6, 83, 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]

(226−77, 226, 240)-Net in Base 4 — Constructive

(149, 226, 240)-net in base 4, using

4 times m-reduction [i] based on (149, 230, 240)-net in base 4, using
- trace code for nets [i] based on (34, 115, 120)-net in base 16, using
  - base change [i] based on digital (11, 92, 120)-net over F₃₂, using
    - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
        a function field by SÃ©mirat [i]

(226−77, 226, 566)-Net over F₄ — Digital

Digital (149, 226, 566)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(226−77, 226, 18355)-Net in Base 4 — Upper bound on s

There is no (149, 226, 18356)-net in base 4, because

1 times m-reduction [i] would yield (149, 225, 18356)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2908 251232 974782 195880 981821 857760 982894 291329 141333 117528 205015 751580 361615 177142 389655 480477 672927 653154 809948 364640 662939 376869 310740 > 4²²⁵ [i]